Question

The maximum slope of the curve $y=\frac{1}{2}{x}^4-5x^3+18x^2-19x$ occurs at the point :

• A. $(2,9)$
• B. $(2,2)$
• C. $(3,\frac{21}{2})$
• D. $(0,0)$

Answer 1

The correct option is B $(2,2)$

$y=\frac{1}{2}{x}^4-5x^3+18x^2-19x$
$\frac{dy}{dx}=2x^3-15x^2+36x-19$
Let $f\left(x\right)=2x^3-15x^2+36x-19$
$f^{\prime }\left(x\right)=6x^2-30x+36=0$
For extrema, $x^2-5x+6=0$
$\Rightarrow x=2,3$

$f^{\prime \prime }\left(x\right)=12x-30$
$f^{\prime \prime }\left(x\right)<0$ for $x=2$
So, at $x=2$, slope is maximum.
$y=8-40+72-38$
$=72-70=2$
Hence, maximum slope occurs at $(2,2).$