Question

Find all critical numbers of the function 7B%5Cfrac%7B1%7D%7B4%7D%20%7D" id="TexFormula1" title="g(x)=x^{\frac{3}{4} }-2x^{\frac{1}{4} }" alt="g(x)=x^{\frac{3}{4} }-2x^{\frac{1}{4} }" align="absmiddle" class="latex-formula">. Show the steps!

Answer 1

Differentiate g using the power rule:

$g'(x) = \dfrac34 x^{\frac34-1} - 2\cdot\dfrac14 x^{\frac14-1}$

$g'(x) = \dfrac34 x^{-\frac14} - \dfrac12 x^{-\frac34}$

$g'(x) = \dfrac14 x^{-\frac34} \left(3 x^{\frac12} - 2\right)$

$g'(x) = \dfrac{3\sqrt x - 2}{4 x^{\frac34}}$

The critical points of g occur where g' is zero or undefined.

We have

$g'(x) = 0 \implies 3\sqrt x - 2 = 0 \implies \sqrt x = \dfrac23 \implies \boxed{x = \dfrac49}$

and the derivative is undefined for

$\dfrac1{g'(x)} = 0 \implies 4x^{\frac34} = 0 \implies \boxed{x=0}$