I was setting a question when I came across a problem. The question was: Suppose I have a function $y=e^{1+\cos(x)+\sqrt{x}}$.
(A) Locate its turning points by taking derivatives and sketch its graph in the interval $[0, 20]$
(B) Given that I randomly choose 2 points on the graph such that their $x$-coordinates are 2 units apart. Find the probability that within that 2 points contains an extremum point (maxima/minima).
I realise part (B) is relatively easy to solve once we have found the extremum points and identify the range of values of the interval that contains the extremum, but my problem is to solve and find its extremum. $$\frac{dy}{dx}=e^{1+\cos(x)+\sqrt{x}}(-\sin(x)+\frac{1}{2\sqrt{x}})=0$$ $$x=\sin^{-1}(\frac{1}{2\sqrt{x}})$$
My problem is that I am unable to solve this equation. Does anyone know how to solve such equations in general without the help of any software?
Thank you!
Generally speaking, equations of the from $$f(x,\sin x)=0$$ where $f$ is a polynomial (even of first degree) can't be solved by algebraic methods.