Question

The maximum slope of the curve $y=-x^3+3x^2+2x-27$ is

• A. $5$
• B. $-5$
• C. $\frac{1}{5}$
• D. None of these

Answer 1

The correct option is A

$5$

Explanation for correct option

Given curve, $y=-x^3+3x^2+2x-27$

Then, Slope$=\frac{dy}{dx}=-3x^2+6x+2$

Let $f\left(x\right)=-3x^2+6x+2$

We know that maximum value occurs at the critical point.

So, the critical point is given as :

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

$$\begin{gathered} \Rightarrow -x+1=0 \\ \Rightarrow x=1 \end{gathered}$$

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

$\Rightarrow f\text{'}\text{'}\left(x\right)=-6<0$

Thus, at $x=1$, the slope is maximum

Value of Slope at $x=1$ is $f\left(x\right)=-3+6+2=5$

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