JEE MAIN - Mathematics (2024 - 29th January Evening Shift - No. 15)
The function $$f(x)=2 x+3(x)^{\frac{2}{3}}, x \in \mathbb{R}$$, has
exactly one point of local minima and no point of local maxima
exactly one point of local maxima and exactly one point of local minima
exactly two points of local maxima and exactly one point of local minima
exactly one point of local maxima and no point of local minima
Explanation
$$\begin{aligned} & f(x)=2 x+3(x)^{\frac{2}{3}} \\ & f^{\prime}(x)=2+2 x^{\frac{-1}{3}} \\ & =2\left(1+\frac{1}{x^{\frac{1}{3}}}\right) \\ & =2\left(\frac{x^{\frac{1}{3}}+1}{x^{\frac{1}{3}}}\right) \end{aligned}$$
_29th_January_Evening_Shift_en_15_1.png)
So, maxima (M) at x = $$-$$1 & minima (m) at x = 0
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