Let for all real .

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let

g(x)=e^{2x}\cos x

for all real

x

. (a) Derive an explicit expression for

g'(x)

. (b) Determine every value of

x

in the interval

[-\pi,\pi]

at which the tangent to the graph of

g

is horizontal. Justify that you have found all such values. (c) For each

x

found in part (b), decide whether the point is a location of a local maximum, a local minimum, or neither for

g

. Support your answer by analyzing the sign of

g'(x)

on intervals between the critical points.