Prove that $\sin^4n/\sqrt n$ is monotonically decreasing

Question

How can we prove that this function

$$f(x) = \frac{\sin^4x}{\sqrt x}$$

is monotonically decreasing? I tried to use usual method using derivative, but it do not give us an answer.

Answer 1

A quick way could be to plot it using a software like R. The function isn't monotonically decreasing.

Answer 2

$f(x) = \frac{\sin^4{x}}{\sqrt{x}}$ is not monotonic decreasing. To see this, evaluate the derivative at $1$.

Answer 3

We have $$f(x) = \dfrac{\sin^4 x}{\sqrt{x}} \geq 0$$ and $$f(\pi) = 0$$ If it were decreasing, we would have $f(x) = 0$ for every $x \geq \pi$, which is clearly not the case.