Answer the following questions about local extrema of differentiable functions.

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

Problem

Answer the following questions about local extrema of differentiable functions. (i) Define a stationary point of a real-valued function

f

. (ii) State the conditions on

f''(c)

that decide whether a stationary point

x=c

is a local maximum or a local minimum. (iii) For the function

g(x)=x^{4}

, show that

x=0

is a stationary point where

g''(0)=0

. Explain why the nature of this point cannot be concluded directly from

g''(0)

and state what you can say about

x=0

for this function.