Is it true that $e^{\sin(3.14)}e^{3.14} \le e^{\sin(3.15)}e^{3.15}$?

Question

I have to determine whether is it true that $$e^{\sin(3.14)}e^{3.14} \le e^{\sin(3.15)}e^{3.15}$$ and whether it is a equality. I even don't know how to begin with it...

Answer 1

Hint: you can use the mean value theorem on $f(x)=\sin x$ so that $f(y)-f(x)=f'(\xi)(y-x)$ for some $x\le \xi\le y$ provided you can bound the derivative suitably.

Answer 2

Since $exp$ is monotonically increasing, this reduces to $$f(3.14) \le f(3.15)$$ where $$f (x) = x + sin(x)$$ So

$$\begin{align} f'(x) & = 1 + cos (x) \\ & \ge 0 \end{align}$$

So ...

Answer 3

Let f(x)=x+sin(x), x = 3.14, y = pi, z = 3.15. Then using the mean value theorem f(z) > f(y) > f(x),- in both cases ">", but not "≥".