Question

If Rolle's theorem for $f\left(x\right)=e^x(\sin x-\cos x)$ is verified on $[\frac{\pi }{4},\frac{5\pi }{4}]$ then the value of $c$ is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{2}$
• C. $\frac{3\pi }{4}$
• D. $\pi$

Answer 1

The correct option is D $\pi$

In calculus, Rolle's theorem essentially states that any real-valued differentiable function that attains equal values at two distinct points must have a stationary point somewhere between them—that is, a point where the first derivative (the slope of the tangent line to the graph of the function) is zero

$f^1\left(c\right)=0$

$\Rightarrow e^x(sinx-cosx)+e^x(cosx+sinx)=0⟹e^x\left(2sinx\right)=0$

$⟹sinx=0$

From $[\frac{\pi }{4},\frac{5\pi }{4}]$ $sinx=0$ if $x=\pi$

$\Rightarrow c=\pi$