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. $\left(3,\frac{21}{2}\right)$
• D. $(0,0)$

Answer 1

The correct option is B

$(2,2)$

Explanation for correct option

Given curve, $y=\frac{1}{2}{x}^4-5x^3+18x^2-19x$

Then, Slope$=\frac{dy}{dx}=2x^3-15x^2+36x-19$

Let $f\left(x\right)=2x^3-15x^2+36x-19$

We know that maximum value occurs at the critical point.

So, the critical point is given as :

$f\text{'}\left(x\right)=6x^2-30x+36=0$

$$\begin{gathered} \Rightarrow x^2-5x+6=0 \\ \Rightarrow (x-2)(x-3)=0 \\ \Rightarrow x=2,3 \end{gathered}$$

We know that for maximum value $f\text{'}\text{'}\left(x\right)<0$

$\Rightarrow f\text{'}\text{'}\left(x\right)=12x–30$

$\Rightarrow f\text{'}\text{'}\left(2\right)=12\left(2\right)–30=-6<0$

Thus, at $x=2$, the slope is maximum

Value of $y$ coordinate at $x=2$ is $y=8–40+72–38=72–70=2$

Thus, maximum slope occurs at $(2,2)$.

Hence, Option $\left(\mathrm{B}\right)$ is the correct answer.