Which of the following statements is/are correct? (A) lim_(x-0) [integral from 0 to x of t*e^(t²) dt] / [1 + x - e^x] equals -2. (B) Points L and M lie on the curve 14x² - 7xy + y² = 2, both with x-coordinate equal to 1. If the tangents to the curve at L and M meet at point (h, k), then k

Question

Which of the following statements is/are correct? (A) lim_(x->0) [integral from 0 to x of t*e^(t²) dt] / [1 + x - e^x] equals -2. (B) Points L and M lie on the curve 14x² - 7xy + y² = 2, both with x-coordinate equal to 1. If the tangents to the curve at L and M meet at point (h, k), then k equals 4. (C) Let f(x) = |x - a1| + |x - a2| +... + |x - an|, where a1 < a2 <... < an are real numbers. If n is even, then f(x) attains its minimum value at exactly one point. (D) If the Mean Value Theorem (MVT) is applicable to a quadratic function y = px² + qx + r on [x1, x2], then the point c guaranteed by MVT satisfies c = (x1 + x2)/3.

Accepted Answer

Points L and M lie on 14x² - 7xy + y² = 2 with x = 1; tangents at L and M meet at (h, k) where k = 4.. (A) L'Hopital once: numerator -> x*e^(x²) -> 0, denominator -> 1-e^x -> 0. Apply again: numerator' = e^(x²)+2x²*e^(x²) -> 1; denominator' = -e^x -> -1. Limit = -1, not -2. FALSE. (B) At x=1: 14-7y+y²=2 => y²-7y+12=0 => y=3 or y=4. L=(1,3), M=(1,4). Implicit differentiation: 28x-7y-7xy'+2yy'=0. At (1,3): 28-21-7y'+6y'=0 => y'=7. Tangent: y-3=7(x-1). At (1,4): 28-28-7y'+8y'=0 => y'=0. Tangent: y=4. Intersection: y=4, then 4-3=7(x-1) => x=8/7. So (h,k)=(8/7,4) and k=4. TRUE. (C) For even n, minimum is attained on the interval [a_(n/2), a_(n/2+1)] (all points in between), not a single point. FA

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