Question

The equation $e^{x-8}+2x-17=0$ has

• A. two real roots
• B. one real root
• C. eight real roots
• D. four real roots

Answer 1

Answer: (B)

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 $f^{\prime }\left(\beta \right)=0$.
But $f^{\prime }\left(\beta \right)=e^{\beta -8}+2$, so that $e^{\beta -8}=-2$ which is not possible, Hence there is no real root other than $8$.