Define (F(x) = (1 + e^ (x^ 2 ) ) )

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Define F(x)=ln⁡(1+esin⁡(x2))F(x) = \ln\bigl(1 + e^{\sin(x^{2})}\bigr), where ln⁡\ln denotes the natural logarithm. All required derivatives exist for real x. (a) Show, without simplifying, an explicit expression for F′(x)F'(x). (b) Using your expression from part (a), compute F′′(x)F''(x) and reorganize terms so that each distinct elementary function appears at most once in every summand. (c) Determine the exact numerical value of F′′(0)F''(0). (d) Interpret the sign of F′′(0)F''(0) in terms of the concavity of FF at x = 0.

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