The number of critical points of the function f ( x ) = ( x - 2 ) 2 / 3 ( 2 x + 1 ) is [2024]

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Solution

Q.
The number of critical points of the function $f (x) = {(x - 2)}^{2 / 3} (2 x + 1)$ is [2024]

1 1

2 2

3 3

4 0

Ans.
(2)

Given, $f (x) = {(x - 2)}^{\frac{2}{3}} (2 x + 1)$

$f' (x) = \frac{2}{3} {(x - 2)}^{- \frac{1}{3}} (2 x + 1) + 2 {(x - 2)}^{\frac{2}{3}}$

$= 2 {(x - 2)}^{- \frac{1}{3}} [\frac{2 x + 1}{3} + (x - 2)]$

$= \frac{2}{{(x - 2)}^{\frac{1}{3}}} [\frac{2 x + 3 x + 1 - 6}{3}] = \frac{2}{{(x - 2)}^{\frac{1}{3}}} \frac{(5 x - 5)}{3}$

$= \frac{10}{3} \frac{(x - 1)}{{(x - 2)}^{\frac{1}{3}}}$

For critical points, $f' (x) = 0$ or $f' (x)$ is non-existence.

$∴$ Critical points are $x = 1, 2$ i.e., 2 in number.

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