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  4. Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2

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Q. Find all the points of local maxima and local minima of the function $f(x) = (x - 1)^3 (x + 1)^2$

Application of Derivatives

Solution:

Let $y = f(x) = (x - 1)^3(x + 1)^2$. Then,
$\frac{dy}{dx} = 3\left(x-1\right)^{2} \left(x + 1\right)^{2} + 2\left(x+ 1\right)\left(x - 1\right)^{3}$
$\Rightarrow \frac{dy}{dx} = \left(x - 1\right)^{2 }\left(x + 1\right)\left\{3\left(x +1\right) + 2\left(x -1\right)\right\}$
$\Rightarrow \frac{dy}{dx} = \left( x - 1\right)^{ 2} \left(x+1\right)\left(5x + 1\right)$
For local maximum or local minimum, we have
$ \frac{dy}{dx} = 0 \Rightarrow \left(x -1\right)^{2}\left(x + 1\right)\left(5x + 1\right) = 0$
$\Rightarrow x = 1$ or, $x = - 1$ or,
$x = -\frac{1}{5}$