Mathematics

Problem

Consider the function

F(x) = e^{-x} + x^{2}

0 \le x \le 4

. (a) Show that

F

is continuous on

[0,4]

and differentiable on

(0,4)

. (b) Find

F'(x)

and prove that there is exactly one critical point

c

(0,4)

. (c) Compute

F''(c)

and classify the critical point. (d) Evaluate

F

x = 0

x = 4

, and

x = c

. Use these values to identify the absolute minimum of

F

[0,4]

and justify your conclusion. (e) Determine the absolute maximum value of

F

on the interval and locate where it occurs.