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Q.
The equation $e^{x-8} + 2x - 17 = 0$ has
Continuity and Differentiability
Solution:
Clearly $x = 8$ satisfies the given equation. Assume that $f(x)=e^{x-8} + 2x - 17 = 0$ has a real root $\alpha$ other than $x = 8$. We may suppose that $\alpha < 8$ (the case for $a < 8$ is exactly similar). Applying Rolle's theorem on $[8, \alpha],$
we get $\beta\in\left(8, \alpha\right)$, such that $f '\left(\beta\right)=0$.
But $f '\left(\beta\right)=e^{\beta-8}+2$, so that $e^{\beta-8}=-2$ which is not possible, Hence there is no real root other than $8$.