Let be a function that is continuous on and differentiable on for every

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

Problem

Let

q

be a function that is continuous on

[0,k]

and differentiable on

(0,k)

for every

k>0

. Suppose the derivative of

q

is always bounded between

-2

and

6

on that interval:

-2\le q'(x)\le 6\quad\text{for }0\le x\le k.

Assume that

q(0)=5

. (1) Express upper and lower bounds for

q(k)

in terms of

k

. Provide complete justification for your bounds. (2) Using the bounds from part (a), determine the greatest integer value of

k

(if any) for which

q(k)\le 25

is guaranteed. Explain your reasoning clearly. (3) Give an explicit numerical example of a value of

k

and a differentiable function

q

(defined on

[0,k]

and satisfying the derivative bounds) that attains your upper bound for

q(k)

from part (a). Justify that your example meets all required conditions.