Question

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

• A. no relative extrema, one point of inflection.
• B. two relative maxima, one relative minimum, two points of inflection.
• C. one relative maximum, one relative minimum, one point of inflection.
• D. one relative maximum, one relative minimum, two point of inflection.

Answer 1

Answer: (C)

$f^{\prime }(x)=\frac{x^4}{4}-\frac{x^3}{3}=\frac{x^3}{12}(3x-4)$
$f^{\prime \prime }(x)=x^3-x^2=x^2(x-1)$
$$\begin{gathered} R\text{ for }x<0,f^{\prime }(x)>0\text{ and } \\ <br/>Rx>0,f^{\prime }(x)<0\text{ hence at }x=0\text{, we have maxima } \end{gathered}$$
$\text{ Also }{f}^{\prime \prime }(0)=0\text{ when }x=1\text{ and }x=0$ $\text{ but at }x=1\text{ only we have an inflection point }$
$\Rightarrow (C)$