Mathematics

Problem

A function

h

is twice differentiable on

(-\infty,\infty)

and is defined by

h(x)=\dfrac{x^{2}}{x^{2}+k}

, where

k

is a positive constant. (a) Compute

h'(x)

and show that

h'(x)=0

when

x=0

regardless of the value of

k

. (b) Find

h''(0)

in terms of

k

. (c) For what values of

k

does the point

x=0

correspond to a local maximum of

h

? For what values of

k

does it correspond to a local minimum? Fully justify your conclusions. (d) Describe how your answer in part (c) would change, if at all, if

k

were allowed to be negative. Provide an explanation without recalculating derivatives.

Intuitive explanation

The key is recognizing that $h(x)$ is a rational function where the numerator and denominator have the same degree, so its behavior is dominated by their leading coefficients as $|x| \to \infty$ . To understand the function's shape—where it increases, decreases, and curves—you'll need to compute $h'(x)$ and $h''(x)$ using the quotient rule, then analyze their signs. This reveals critical points, monotonicity, and concavity, which together paint a complete picture of how this function behaves.

Intuitive explanation

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