A differentiable function is defined on the open interval by h(x)=x^ 3 -6x^ 2 +9x
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A differentiable function h is defined on the open interval (0,4) by
h(x)=x3−6x2+9x.
Use the information below to analyze the extrema of h and to explore how the interval on which a function is defined influences the existence of absolute extrema.
(a) Describe the difference between a local (relative) extremum and an absolute (global) extremum of a function.
(b) Determine every local maximum and local minimum value of h on the open interval (0,4) and the x–values at which they occur.
(c) Does h attain an absolute maximum value or an absolute minimum value on (0,4)? Provide a complete explanation that references both the results of part (b) and the conditions required by the Extreme Value Theorem.