The function f ( x ) = 2 x + 3 ( x ) 2 / 3 , x ? R , has [2024],

Question

The function $f (x) = 2 x + 3 {(x)}^{2 / 3}, x \in R,$ has [2024]

Smart Achievers · Accepted Answer

(4)

We have, $f (x) = 2 x + 3 x^{2 / 3}$

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

For critical points, $f' (x) = 0$

$\Rightarrow 1 + x^{- 1 / 3} = 0 \Rightarrow x^{1 / 3} = - 1 \Rightarrow x = - 1$

The sign scheme of $f' (x)$ is

$∴$ $x = - 1 is the point of local maxima and x = 0 is the point of local minima.$

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