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.
Asked
Active
Viewed 303 times
-1
-
2It is not monotonically decreasing. – Kay K. Dec 31 '15 at 14:58
-
it must be $$f(x)=\frac{\sin(x)^4 }{\sqrt{x}}$$ – Dr. Sonnhard Graubner Dec 31 '15 at 14:59
-
see here http://www.wolframalpha.com/input/?i=plot+sin%28x%29%5E4%2Fsqrt%28x%29+for+x%3D0+to+10 the given function isn't monoton – Dr. Sonnhard Graubner Dec 31 '15 at 15:03
-
Try $g(x)=f(x^2)$. – DanielWainfleet Dec 31 '15 at 17:46
3 Answers
3
A quick way could be to plot it using a software like R. The function isn't monotonically decreasing.
Amey Joshi
- 1,084
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.
chi
- 2,143
2
$f(x) = \frac{\sin^4{x}}{\sqrt{x}}$ is not monotonic decreasing. To see this, evaluate the derivative at $1$.
fosho
- 6,334