Practicing Success
The number of solutions of the equation $x\,e^{\sin x} - \cos x = 0$ in the interval (0, π/2), is _____. |
1 |
Let $f (x) = x\, e^{\sin x} - cos x$. Then, $f'(x)=e^{\sin x}+x\, e^{\sin x} \cos x + \sin x > 0$ for all $x ∈ (0, π/2)$ ⇒ f(x) is increasing on (0, π/2) Also, $f (0) = -1$ and $f(\frac{π}{2})=\frac{π}{2}e$ So, there is a point between 0 and π/2, where $f(x) = 0$. Hence, $f(x) = 0$ has exactly one solution in (0, π/2). |