The function f(x) = 2x + 3(x)^2/3, x ∈ R, has

Question

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

Answer 1

(1) exactly one point of local minima and no point of local maxima

\(f(x)=2 x+3(x)^{\frac{2}{3}}\)

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

\(=2\left(1+\frac{1}{x^{\frac{1}{3}}}\right)\)

\(=2\left(\frac{x^{\frac{1}{3}}+1}{x^{\frac{1}{3}}}\right)\)

So, maxima (M) at x = -1 & minima (m) at x = 0.