Consider the one–parameter family of functions defined on the closed interval by…
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
Consider the one–parameter family of functions fp defined on the closed interval [0,5] by
fp(x)=x3−3px2+2,
where p is a real constant.
For the specific value p=2, answer the following questions about f2 on [0,5].
(a) Determine all critical points of f2 that lie in the interval [0,5]. Show analytical work that leads to their x–coordinates.
(b) Evaluate f2 at each critical point found in part (a) and at the endpoints x=0 and x=5.
(c) Using the values obtained in part (b), state the absolute maximum value of f2 on [0,5], the x–value(s) where it occurs, the absolute minimum value, and where it occurs. Justify your conclusion.