Let , where is a real parameter. The function is defined on the set of real numbers…

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

Problem

Let

h(x)=\dfrac{x^{2}}{x^{2}+kx+1}

, where

k

is a real parameter. The function is defined on the set of real numbers for which the denominator is non-zero. (a) Find the domain of

h

in terms of

k

. (b) Obtain a simplified formula for

h''(x)

as a rational function of

x

and

k

. (c) Determine all real values of

k

for which

h

is concave down on

(0,\infty)

. Show all algebraic steps leading to your conclusion. (d) For each value of

k

identified in part (c), find the

x

-coordinate(s) of any inflection point(s) of

h

that lie in its domain.