<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] Let
$$y = x^{\frac{1}{3}}$$
So,
$$\frac{5}{3}y^{2} - \frac{4k}{3}y + k^{2} > 0 \Longleftrightarrow 5y^{2} - 4ky + 3k^{2} > 0$$
and
$$(-4k)^{2} - 4.5.3k^{2} = 16k^{2} - 60k^{2} < 0.$$
Therefore, $\frac{5}{3}y^{2} - \frac{4k}{3}y + k^{2} > 0$. Complete the solution.
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