Mathematics

Problem

A quadratic function is given by

p(x)=x^{2}-4kx

, where

k

is a real parameter. Consider the closed interval

[k,k+2]

. a. Express the average rate of change of

p

[k,k+2]

as a function of

k

. Simplify completely. b. Find all real values of

k

for which the average rate of change on

[k,k+2]

equals

8

. c. For each value of

k

obtained in part (b), determine the sign (positive, negative, or zero) of the average rate of change if

k

is perturbed slightly upward. Explain your reasoning without recomputing the entire expression.

Intuitive explanation

The key is recognizing that a parabola's minimum on a closed interval depends on where its vertex sits relative to that interval. Since $p(x) = x^2 - 4kx$ has vertex at $x = 2k$ , you need to check whether $2k$ falls inside, to the left of, or to the right of $[k, k+2]$ — each case gives a different location for the minimum value. This parameter-dependent analysis is what makes the problem interesting.

Intuitive explanation

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