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# Find the Maximum Slope of the Curve Y= − X 3 + 3 X 2 + 2 X − 27 . - CBSE (Science) Class 12 - Mathematics

#### Question

Find the maximum slope of the curve y = $- x^3 + 3 x^2 + 2x - 27 .$

#### Solution

$\text { Given: } \hspace{0.167em} y = - x^3 + 3 x^2 + 2x - 27 ............\left( 1 \right)$

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

$\text { Now,}$

$M = - 3 x^2 + 6x + 2$

$\Rightarrow \frac{dM}{dx} = - 6x + 6$

$\text { For maximum or minimum values of M, we must have }$

$\frac{dM}{dx} = 0$

$\Rightarrow - 6x + 6 = 0$

$\Rightarrow 6x = 6$

$\Rightarrow x = 1$

$\text { Substituing the value of x in eq. } \left( 1 \right),\text { we get }$

$y = - 1^3 + 3 \times 1^2 + 2 \times 1 - 27 = - 23$

$\frac{d^2 M}{d x^2} = - 6 < 0$

$\text { So, the slope is maximum when x = 1 and y } = - 23 .$

$\therefore At \left( 1, - 23 \right):$

$\text { Maximum slope } = - 3 \left( 1 \right)^2 + 6\left( 1 \right) + 2 = - 3 + 6 + 2 = 5$

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Solution Find the Maximum Slope of the Curve Y= − X 3 + 3 X 2 + 2 X − 27 . Concept: Graph of Maxima and Minima.
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