Between any two real roots of the equation e^x sin x = 1, the equation e^x co...

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Question Id : 11530 | Context :NIMCET 2023

Between any two real roots of the equation

$e^x sin x = 1$ , the equation

$e^x cos x = –1$ has

🎥 Video solution / Text Solution of this question is given below:

Given:

$e^x \sin x = 1$ has two real roots → say $x_1$ and $x_2$

Apply Rolle’s Theorem:

Since $f(x) = e^x \sin x$ is continuous and differentiable, and $f(x_1) = f(x_2)$ , ⇒ There exists $c \in (x_1, x_2)$ such that $f'(c) = 0$

Compute:

$f'(x) = e^x(\sin x + \cos x) = 0 \Rightarrow \tan x = -1$ At this point, $e^x \cos x = -1$

$\boxed{\text{At least one root}}$

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