question_answer

The graph of function \[f(x)=\frac{{{x}^{5}}}{20}-\frac{{{x}^{4}}}{12}+5\] has

A)  no local extremum, one point of inflection.

B)  two local maximum, one local minimum, two point of inflection.

C)  one local maximum, one local minimum, one point of inflection.

D)  one local maximum, one local minimum, two point of inflection.

Correct Answer: C

Solution :

\[f(x)=\frac{{{x}^{5}}}{20}-\frac{{{x}^{4}}}{12}+5\] \[f'(x)=\frac{{{x}^{4}}}{4}-\frac{{{x}^{3}}}{3}=\frac{{{x}^{3}}}{12}(3x-4)\] \[\Rightarrow f''(x)={{x}^{3}}-{{x}^{2}}={{x}^{2}}(x-1)\] Now, verify alternative