Question

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

• A. $1$, $-1$, $-1/5$
• B. $1$, $-1$
• C. $1$, $-1/5$
• D. $-1$, $-1/5$

Answer 1

Answer: (A)

$1$, $-1$, $-1/5$

Answer 2

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, $x = -\frac{1}{5}$