Differentiate g using the power rule:
[tex]g'(x) = \dfrac34 x^{\frac34-1} - 2\cdot\dfrac14 x^{\frac14-1}[/tex]
[tex]g'(x) = \dfrac34 x^{-\frac14} - \dfrac12 x^{-\frac34}[/tex]
[tex]g'(x) = \dfrac14 x^{-\frac34} \left(3 x^{\frac12} - 2\right)[/tex]
[tex]g'(x) = \dfrac{3\sqrt x - 2}{4 x^{\frac34}}[/tex]
The critical points of g occur where g' is zero or undefined.
We have
[tex]g'(x) = 0 \implies 3\sqrt x - 2 = 0 \implies \sqrt x = \dfrac23 \implies \boxed{x = \dfrac49}[/tex]
and the derivative is undefined for
[tex]\dfrac1{g'(x)} = 0 \implies 4x^{\frac34} = 0 \implies \boxed{x=0}[/tex]
