Application Of Derivatives Question 3
Question 3 - 2024 (29 Jan Shift 2)
The function $f(x)=2 x+3(x)^{\frac{2}{3}}, x \in \mathbb{R}$, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima
Show Answer
Answer (3)
Solution
$$ \begin{aligned} & f(x)=2 x+3(x)^{\frac{2}{3}} \ & f^{\prime}(x)=2+2 x^{\frac{-1}{3}} \ & =2\left(1+\frac{1}{x^{\frac{1}{3}}}\right) \ & =2\left(\frac{x^{\frac{1}{3}}+1}{x^{\frac{1}{3}}}\right) \end{aligned} $$
So, $\operatorname{maxima}(\mathrm{M})$ at $\mathrm{x}=-1 & \operatorname{minima}(\mathrm{m})$ at $\mathrm{x}=0$
