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