Let be the function defined on by .

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

Problem

Let

p

be the function defined on

[-3,3]

p(x)=|x|+\dfrac{x^{2}}{4}

. (a) Verify that

p

is continuous on the interval

[-3,3]

. (b) Determine

p'(x)

for

x\neq 0

. Discuss the behavior of

p

x=0

and decide whether

x=0

must be considered when searching for extreme values. (c) Solve

p'(x)=0

for

x

(-3,3)

and compile a complete list of interior points that must be tested for absolute extrema. (d) Compute

p(x)

at every point from part (c) and at the endpoints

x=-3

and

x=3

. (e) Using the results from part (d), determine the absolute maximum and absolute minimum of

p

[-3,3]

. Provide full justification, including a discussion of why no other points can produce larger or smaller values on the interval.