How do you find the critical points for $f(x)= x^(4/3) + 4x^(1/3)+ 4x^(-2/3)$ ?

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Answer 1 · 2015-09-25T20:48:31

The critical points are $(1,f(1))=(1,9)$

First the domain of f(x) is $D(f)=(0,+oo)$ hence

we must find the roots of the first derivative which is

$(df(x))/dx=(4*(x^2+x-2))/(3x^(5/3))$

The roots of the derivative are the roots of the equation $x^2+x-2=0$ hence $x=-2$ , $x=1$ .

But $x=-2$ does mot belong in the domain of f(x).

How do you find the critical points for f(x)= x^(4/3) + 4x^(1/3)+ 4x^(-2/3)f(x)= x^(4/3) + 4x^(1/3)+ 4x^(-2/3)?