the greatest value of an implicit function $$\frac{\exp(x)\sin(x) - \exp(y)\sin(y)}{\exp(y)-\exp(x)} \le P$$ where $x \le y$
how would you find the smallest value of $P$ - the greatest value of the LHS implicit function without using a graphical calculator
In addition $x$ is less than or equal to $y$ that means the graph must lie on the LHS of the graph $y=x$

By setting $u=\exp x, v=\exp y$ and $f(w)=-w\sin\log w$ we are looking for
$$P=\sup_{0<u<v}\frac{f(v)-f(u)}{v-u}$$
but the Lagrange theorem gives:
$$\frac{f(v)-f(u)}{v-u}=f'(\xi),\qquad \xi\in(u,v)$$
and since:
$$ f'(w) = -\sin(\log w)-\cos(\log w)=-\sqrt{2}\sin(\log w+\pi/4)$$
we have $\color{red}{P=\sqrt{2}}$.
