<s>[INST] Show that $f(x)=x^{5/3}-kx^{4/3}+k^2x$ is increasing for $k\neq0$ So to show the function is increasing/decreasing we differentiate and show it is more than zero/less than zero:
We have
$$f(x)=x^{5/3}-kx^{4/3}+k^2x$$
Hence,
$$f'(x)=\frac{5}{3}x^{2/3}-\frac{4k}{3}x^{1/3}+k^2$$
But how do I show
$$\frac{5}{3}x^{\frac{2}{3}}-\frac{4k}{3}x^{\frac{1}{3}}+k^2>0$$
 [/INST] Notice that you now have a new second order polynomial equation with $k$ as a variable instead of x.
You can compute the discriminant $\Delta$:
$$\Delta = (\frac{4x^{\frac{1}{3}}}{3})^2 - 4\frac{5x^\frac{2}{3}}{3} $$
$$\Delta = \frac{16x^{\frac{2}{3}}}{9} - \frac{60x^\frac{2}{3}}{9} = -\frac{44x^\frac{2}{3}}{9} < 0$$
Therefore $f'$ has no root and has the same sign as $k^2$
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