💡Alright, so I have this problem here: "The negation of the proposition 'For all x in ℝ, x² - x + 2 ≥ 0' is." Okay, so I need to find the negation of this statement. Hmm, let's break it down step by step.First, the original statement is saying that for every real number x, when you plug it into the quadratic expression x² - x + 2, the result is always greater than or equal to zero. So, it's a universal statement—it's making a claim about all real numbers x.Now, to find the negation of this statement, I remember that the negation of a universal quantifier ("for all") is an existential quantifier ("there exists"). So, instead of saying "for all x," the negation would say "there exists an x" such that the original condition is not true. In logical terms, ¬(∀x P(x)) is equivalent to ∃x ¬P(x).Applying that to this problem, the negation should be: "There exists an x in ℝ such that x² - x + 2 is not greater than or equal to zero." Which simplifies to "There exists an x in ℝ such that x² - x + 2 < 0."Wait, but is that all? Let me make sure I'm not missing anything. The original statement is about the quadratic expression x² - x + 2 being non-negative for all real x. So, the negation would be that there's at least one real x where this expression is negative.But hold on, I should probably check if such an x actually exists. Maybe the quadratic is always positive, which would mean the negation is false, but that's a different question. The problem is just asking for the negation, not whether it's true or false.So, focusing back on the negation: starting with "for all x in ℝ, x² - x + 2 ≥ 0," the negation is indeed "there exists an x in ℝ such that x² - x + 2 < 0." That seems straightforward.Let me think if there's another way to express this. Maybe using logical symbols, it would be ∃x ∈ ℝ, x² - x + 2 < 0. Yeah, that's concise and clear.Is there anything else I need to consider? Maybe the domain of x? The original statement specifies x in ℝ, so the negation should maintain that domain. So, it's still about real numbers x.Also, just to double-check, the quadratic x² - x + 2. Its discriminant is b² - 4ac = (-1)² - 4*1*2 = 1 - 8 = -7. Since the discriminant is negative, the quadratic doesn't cross the x-axis and is always positive because the coefficient of x² is positive. So, in reality, the original statement is true, and its negation is false. But again, the question is just about forming the negation, not evaluating its truth.So, to recap: The original statement is a universal claim about all real numbers x. The negation flips the universal quantifier to an existential one and negates the inequality. Therefore, the negation is "There exists an x in ℝ such that x² - x + 2 < 0."I think that's solid. I don't see any mistakes in my reasoning. The key steps were recognizing the structure of the original statement, applying the rule for negating universal quantifiers, and ensuring the domain and inequality were correctly addressed in the negation.

💡First, I need to determine the slope of the line passing through points ( M(m, 3) ) and ( N(1, m) ). The formula for the slope between two points ( (x_1, y_1) ) and ( (x_2, y_2) ) is:[k_{MN} = frac{y_2 - y_1}{x_2 - x_1}]Substituting the given points:[k_{MN} = frac{m - 3}{1 - m}]Since the line is perpendicular to another line with a slope of (-1), the product of their slopes must be (-1):[k_{MN} times (-1) = -1]Substituting the expression for ( k_{MN} ):[frac{m - 3}{1 - m} times (-1) = -1]Simplifying:[frac{m - 3}{1 - m} = 1]Solving for ( m ):[m - 3 = 1 - m][2m = 4][m = 2]Therefore, the value of ( m ) is **2**.

💡Okay, so I have this proposition: "The distance from a point on the perpendicular bisector of a line segment to the two endpoints of the line segment is equal." I need to rewrite this in the form "If ..., then ...".First, I need to understand what the proposition is saying. It's talking about a point that's on the perpendicular bisector of a line segment. The perpendicular bisector is a line that cuts a line segment into two equal parts at 90 degrees. So, any point on this perpendicular bisector has some special property related to the endpoints of the line segment.The proposition says that the distance from such a point to the two endpoints is equal. So, if I pick any point on the perpendicular bisector, and measure its distance to both ends of the line segment, those distances will be the same.Now, to rewrite this in the "If ..., then ..." form, I need to identify the condition and the result. The condition is that the point is on the perpendicular bisector, and the result is that the distances to the endpoints are equal.So, putting it together: If a point is on the perpendicular bisector of a line segment, then the distances from that point to the two endpoints of the line segment are equal.Wait, is that all? Maybe I should double-check. The original statement is about the distance from a point on the perpendicular bisector being equal to the two endpoints. So, yes, the condition is being on the perpendicular bisector, and the result is the equality of distances.I think that's correct. But maybe I should think about it differently. Suppose I have a line segment AB, and its perpendicular bisector. If I take any point P on this bisector, then PA equals PB. That's what the proposition is saying.So, in the "If ..., then ..." form, it's like: If P is on the perpendicular bisector of AB, then PA equals PB.Yeah, that makes sense. So, the general form would be: If a point lies on the perpendicular bisector of a line segment, then the distances from that point to the two endpoints of the line segment are equal.I think that's the correct way to rephrase it. It clearly states the condition (being on the perpendicular bisector) and the result (equal distances to the endpoints). I don't see any issues with this phrasing. It accurately captures the original proposition in the desired logical structure.

💡Okay, so I have this calculus problem here: Given y equals e raised to the power of 3x minus cosine of x, and I need to find y prime, which is the derivative of y with respect to x. The options are A, B, C, and D, but D isn't shown here. Let me try to figure this out step by step.First, I remember that to find the derivative of a function, I can differentiate each term separately. So, I have two terms here: e^{3x} and -cos x. I'll need to find the derivative of each one and then combine them.Starting with the first term, e^{3x}. I recall that the derivative of e^{kx} with respect to x is k*e^{kx}. So, in this case, k is 3. That means the derivative of e^{3x} should be 3*e^{3x}. Okay, that seems straightforward.Now, moving on to the second term, which is -cos x. The derivative of cos x with respect to x is -sin x. But since there's a negative sign in front of the cosine, the derivative becomes -(-sin x), which simplifies to sin x. So, the derivative of -cos x is sin x.Putting it all together, the derivative of y should be the derivative of the first term plus the derivative of the second term. That would be 3*e^{3x} + sin x.Wait, let me double-check that. So, derivative of e^{3x} is 3e^{3x}, and derivative of -cos x is sin x. So, adding them together, y' = 3e^{3x} + sin x. Hmm, that seems right.Looking back at the options provided:A: e^{3x} - cos xB: (3 + sin x)e^{3x} - cos xC: e^3 + sin xD: [Not shown]Hmm, none of these options exactly match what I got. Wait, let me look again. Option B is (3 + sin x)e^{3x} - cos x. That doesn't look like what I have. My result was 3e^{3x} + sin x. Option B seems to have an extra term, -cos x, which wasn't in my derivative.Wait, maybe I made a mistake. Let me go through the differentiation again. The function is y = e^{3x} - cos x. So, derivative of e^{3x} is 3e^{3x}, and derivative of -cos x is sin x. So, y' = 3e^{3x} + sin x. That's correct.Looking at the options, none of them seem to match exactly. Option A is the original function, so that's not the derivative. Option B has (3 + sin x)e^{3x} - cos x, which is different from my result. Option C is e^3 + sin x, which doesn't make sense because e^3 is a constant, and the derivative should have an x term. So, maybe I'm missing something here.Wait, perhaps I misread the options. Let me check again. Option B is (3 + sin x)e^{3x} - cos x. That would mean they multiplied (3 + sin x) with e^{3x}, which isn't what I did. I just added 3e^{3x} and sin x. So, maybe the correct answer isn't listed here, or perhaps I made a mistake in my differentiation.Alternatively, maybe the question had a typo or the options are incorrect. But assuming everything is correct, my derivative is 3e^{3x} + sin x, which isn't among the options. Wait, perhaps I should consider that the derivative of -cos x is sin x, so maybe the options have a different arrangement.Wait, let me think again. Maybe I should write it as 3e^{3x} + sin x, which can be factored as e^{3x}(3) + sin x. But none of the options present it that way. Option B is (3 + sin x)e^{3x} - cos x, which would expand to 3e^{3x} + sin x e^{3x} - cos x. That's different from my result because I have 3e^{3x} + sin x, not 3e^{3x} + sin x e^{3x} - cos x.Hmm, maybe I need to reconsider. Perhaps the original function was different? Let me check again: y = e^{3x} - cos x. Yes, that's correct. So, the derivative should be 3e^{3x} + sin x.Wait, maybe the options are written differently. Let me see if any of them can be simplified to my result. Option B: (3 + sin x)e^{3x} - cos x. If I expand that, it's 3e^{3x} + sin x e^{3x} - cos x. That's not the same as my derivative. So, perhaps the correct answer isn't listed, or I made a mistake.Alternatively, maybe I should check if I applied the chain rule correctly. For e^{3x}, the derivative is 3e^{3x}, which I did. For -cos x, the derivative is sin x, which I also did. So, I think my differentiation is correct.Wait, perhaps the options are mislabeled or there's a typo. Alternatively, maybe I need to consider that the derivative of -cos x is sin x, but perhaps the negative sign was misplaced. Let me check again: derivative of -cos x is sin x, because derivative of cos x is -sin x, so negative of that is sin x. So, that part is correct.Hmm, I'm confused because my result doesn't match any of the options. Maybe I should look for a different approach or see if I can manipulate my result to match one of the options.Alternatively, perhaps the original function was y = e^{3x} - cos x, and the derivative is y' = 3e^{3x} + sin x. So, if I factor out e^{3x}, it would be e^{3x}(3) + sin x, but that doesn't help much. Alternatively, maybe I can write it as 3e^{3x} + sin x, which is the same as my result.Wait, looking back at the options, option B is (3 + sin x)e^{3x} - cos x. That seems to have an extra -cos x term, which wasn't in my derivative. So, perhaps the correct answer isn't listed, or maybe I made a mistake in reading the problem.Alternatively, maybe the original function was y = e^{3x} - cos x, and the derivative is y' = 3e^{3x} + sin x, which isn't among the options. So, perhaps the correct answer is not listed, or there's a typo in the options.Wait, maybe I should consider that the derivative of e^{3x} is 3e^{3x}, and the derivative of -cos x is sin x, so y' = 3e^{3x} + sin x. That's my final answer. So, perhaps the correct option is not listed, or maybe I need to choose the closest one.Alternatively, maybe I made a mistake in the differentiation. Let me try differentiating again. For e^{3x}, the derivative is 3e^{3x}. For -cos x, the derivative is sin x. So, y' = 3e^{3x} + sin x. Yes, that's correct.Wait, perhaps the options are written in a different form. Let me see if I can manipulate my result to match any of the options. My result is 3e^{3x} + sin x. Option B is (3 + sin x)e^{3x} - cos x. If I expand option B, it's 3e^{3x} + sin x e^{3x} - cos x. That's different from my result because I have 3e^{3x} + sin x, not involving e^{3x} multiplied by sin x or subtracting cos x.So, I think my result is correct, and perhaps the options are incorrect or there's a typo. Alternatively, maybe I misread the original function. Let me check again: y = e^{3x} - cos x. Yes, that's correct.Wait, perhaps the original function was y = e^{3x} - cos x, and the derivative is y' = 3e^{3x} + sin x. So, if I have to choose from the given options, none of them match exactly. But perhaps the closest one is option B, but it's not correct because it includes an extra term.Alternatively, maybe I should consider that the derivative of -cos x is sin x, so perhaps the negative sign was misplaced in the options. Let me check option B again: (3 + sin x)e^{3x} - cos x. If I remove the -cos x, it would be closer to my result, but it's still not the same.Hmm, I'm stuck. My differentiation seems correct, but none of the options match. Maybe I should go with the closest one, but I'm not sure. Alternatively, perhaps the correct answer is not listed, and I should indicate that.Wait, perhaps I made a mistake in the differentiation. Let me try once more. The function is y = e^{3x} - cos x. The derivative of e^{3x} is 3e^{3x}, and the derivative of -cos x is sin x. So, y' = 3e^{3x} + sin x. Yes, that's correct.Therefore, I think the correct answer is y' = 3e^{3x} + sin x, which isn't listed among the options. So, perhaps the correct answer is not provided, or there's a mistake in the options.Alternatively, maybe the original function was different, and I misread it. Let me check again: y = e^{3x} - cos x. Yes, that's correct.Wait, perhaps the options are written in a different form. Let me see if I can factor my result differently. My result is 3e^{3x} + sin x. I can't factor e^{3x} out because sin x isn't multiplied by e^{3x}. So, it's just 3e^{3x} + sin x.Looking back at the options, none of them present it that way. So, I think my conclusion is that the correct answer isn't listed, or there's a typo in the options.But since I have to choose from the given options, perhaps I should reconsider. Maybe I made a mistake in the differentiation. Let me try again.Derivative of e^{3x} is 3e^{3x}, correct. Derivative of -cos x is sin x, correct. So, y' = 3e^{3x} + sin x. That's correct.Wait, maybe the options are written with a different grouping. Let me see: Option B is (3 + sin x)e^{3x} - cos x. If I expand that, it's 3e^{3x} + sin x e^{3x} - cos x. That's different from my result because I have 3e^{3x} + sin x, not 3e^{3x} + sin x e^{3x} - cos x.So, I think my result is correct, and the options are incorrect. Therefore, I should indicate that none of the options are correct, or perhaps there's a mistake in the problem.But since I have to choose from the given options, and my result doesn't match any, I might have to choose the closest one, but I'm not sure. Alternatively, maybe I made a mistake in the differentiation.Wait, perhaps I should consider that the derivative of e^{3x} is 3e^{3x}, and the derivative of -cos x is sin x, so y' = 3e^{3x} + sin x. That's correct.Therefore, I think the correct answer is y' = 3e^{3x} + sin x, which isn't listed among the options. So, perhaps the correct answer is not provided, or there's a mistake in the options.In conclusion, after carefully differentiating each term and verifying my steps, I believe the correct derivative is 3e^{3x} + sin x, which doesn't match any of the given options. Therefore, there might be an error in the problem or the options provided.

💡First, I recognize that the logarithm base 10 of 100, written as (lg 100), is being asked.Next, I recall that logarithms answer the question: "To what power must the base be raised to obtain the given number?" In this case, the base is 10, and the number is 100.I know that (10) raised to the power of (2) equals (100), because (10 times 10 = 100).Therefore, (lg 100) is equal to (2).

💡Okay, so I have this trigonometry problem here: Given that (cos α + sin α)/(cos α - sin α) = 3/5, I need to find the value of cos²α - sin²α. Hmm, let me think about how to approach this.First, I remember that expressions involving cos α and sin α can sometimes be simplified by dividing both the numerator and the denominator by cos α. That might help me express everything in terms of tan α, which could make things easier. Let me try that.So, if I divide both the numerator and the denominator by cos α, the expression becomes:(cos α + sin α)/(cos α - sin α) = (1 + tan α)/(1 - tan α)And according to the problem, this equals 3/5. So, I can set up the equation:(1 + tan α)/(1 - tan α) = 3/5Now, I need to solve for tan α. Let me cross-multiply to get rid of the fractions:5(1 + tan α) = 3(1 - tan α)Expanding both sides:5 + 5 tan α = 3 - 3 tan αNow, let's collect like terms. I'll bring all the tan α terms to one side and the constants to the other:5 tan α + 3 tan α = 3 - 5Combining like terms:8 tan α = -2So, tan α = -2/8 = -1/4Alright, so tan α is -1/4. Now, I need to find cos²α - sin²α. I recall that cos²α - sin²α is equal to cos(2α), but maybe I can express this in terms of tan α as well.Another approach is to express cos²α - sin²α in terms of tan α. Let me think. Since tan α = sin α / cos α, I can express sin α in terms of tan α and cos α, but that might complicate things. Alternatively, I can use the identity:cos²α - sin²α = (cos²α - sin²α)/(cos²α + sin²α) * (cos²α + sin²α)But since cos²α + sin²α = 1, this simplifies to:cos²α - sin²α = (cos²α - sin²α)/1 = cos²α - sin²αWait, that doesn't help. Maybe I should express cos²α and sin²α in terms of tan α.I know that:cos²α = 1 / (1 + tan²α)sin²α = tan²α / (1 + tan²α)So, cos²α - sin²α = [1 / (1 + tan²α)] - [tan²α / (1 + tan²α)] = (1 - tan²α)/(1 + tan²α)Ah, that's a useful identity. So, cos²α - sin²α = (1 - tan²α)/(1 + tan²α)Since I already found that tan α = -1/4, let's plug that in:tan²α = (-1/4)² = 1/16So, plugging into the expression:(1 - 1/16)/(1 + 1/16) = (15/16)/(17/16) = (15/16) * (16/17) = 15/17So, cos²α - sin²α = 15/17Let me double-check my steps to make sure I didn't make any mistakes.1. I started by dividing numerator and denominator by cos α, which is valid as long as cos α ≠ 0. Since the original expression is given as 3/5, which is a finite number, cos α can't be zero because that would make the denominator undefined. So, that step is fine.2. Then, I set up the equation (1 + tan α)/(1 - tan α) = 3/5 and solved for tan α. Cross-multiplying gave me 5(1 + tan α) = 3(1 - tan α). Expanding both sides: 5 + 5 tan α = 3 - 3 tan α. Bringing like terms together: 5 tan α + 3 tan α = 3 - 5, which simplifies to 8 tan α = -2, so tan α = -1/4. That seems correct.3. Then, I used the identity cos²α - sin²α = (1 - tan²α)/(1 + tan²α). Plugging in tan α = -1/4, I calculated tan²α = 1/16. So, 1 - tan²α = 1 - 1/16 = 15/16, and 1 + tan²α = 1 + 1/16 = 17/16. Dividing these gives (15/16)/(17/16) = 15/17. That seems right.I think all the steps are correct. So, the value of cos²α - sin²α is 15/17.**Final Answer**The value of cos^{2}alpha - sin^{2}alpha is boxed{dfrac{15}{17}}.