Have the following
$f(x)=x^2 \exp(\sin(x))-\cos(x)$ on the interval $[0,\pi/2]$, I have shown the function is continuous and that there is at least one solution on the interval via using IVT, I know I have to find another solution in the interval such that $f(x)<0$, $f(x)>0$ and then $f(x)<0$, just wondering what is the best way to approach, Im sure there must be a more effective way than just to number crunch, could we consider MVT on the interval, many thanks in advance.
Best Answer
The function $x^2exp(sin(x))$ is increasing, the function $cos(x)$ is decreasing in your interval. Hence the combined functions is increasing on your interval. It starts with a value of $-1$ and goes up to $(\pi/2)^2e$. Therefore it has precisely one zero point.