What is the extreme (2x-5)/(3\sqrt[3]{x)}+x^{2/3} ?

The extreme (2x-5)/(3\sqrt[3]{x)}+x^{2/3} is Minimum(-1/2 ,2*2^{1/3}+(-1/2)^{2/3})

extreme (2x-5)/(3\sqrt[3]{x)}+x^{2/3}

extreme points

\frac{2 x - 5}{3 3 x} + x^{\frac{2}{3}}

Minimum (- \frac{1}{2}, 2 \cdot 2^{\frac{1}{3}} + (- \frac{1}{2})^{\frac{2}{3}})

Solution steps

Find the critical points:

x = - \frac{1}{2}

Domain of

\frac{2 x - 5}{3 3 x} + x^{\frac{2}{3}} : x < 0 or x > 0

Find intervals:

Decreasing

: - \infty < x < - \frac{1}{2},

Increasing

: - \frac{1}{2} < x < 0,

Increasing

: 0 < x < \infty

Plug the extreme point

x = - \frac{1}{2}

into

\frac{2 x - 5}{3 3 x} + x^{\frac{2}{3}} \Rightarrow y = 2 2^{\frac{1}{3}} + (- \frac{1}{2})^{\frac{2}{3}}

Minimum (- \frac{1}{2}, 2 \cdot 2^{\frac{1}{3}} + (- \frac{1}{2})^{\frac{2}{3}})

f (x) = 2 (csc (x) + sec (x))

f (x, y) = x e^{y} - ln (x)

y = \frac{5}{x}

f (x) = x^{4} - x^{2} + 2

f (x) = \frac{x ^{2} - 16}{x - 5}

What is the extreme (2x-5)/(3\sqrt[3]{x)}+x^{2/3} ?
The extreme (2x-5)/(3\sqrt[3]{x)}+x^{2/3} is Minimum(-1/2 ,2*2^{1/3}+(-1/2)^{2/3})