Mathematics

Problem

Consider the function

h(x)=x^{3/2}-x^{1/2}

defined for

x\ge 0

. (a) Determine all values of

x

in the interval

[0,9]

for which the tangent line to the graph of

h

is horizontal. (b) For each value of

x

found in part (a), write the equation of the corresponding tangent line. (c) For each critical point identified in part (a), classify

h

as having a local maximum, a local minimum, or neither. Justify your classification by analyzing the sign of

h'(x)

on intervals determined by the critical points.