Consider the function defined on the closed interval by .

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

Problem

Consider the function

g

defined on the closed interval

[-1,1]

g(x)=e^{2x}+x

. (1) Justify that

g

meets all hypotheses required to guarantee at least one point

c

(-1,1)

satisfying

g'(c)=\dfrac{g(1)-g(-1)}{1-(-1)}

. Clearly state each hypothesis and how it is satisfied. (2) Compute the average rate of change of

g

[-1,1]

and express your answer exactly. (3) Solve

g'(c)=\dfrac{g(1)-g(-1)}{1-(-1)}

for all

c

(-1,1)

. Give exact values when possible or justify any numerical approximation used. (4) Identify which of the

c

-values obtained in part (c) also satisfy the additional condition

c>0

, and state your final answer(s).