How to prove Number of solutions to $e^{-x}=x^2-5x+1$ is $3$ without Graphical approach

Question

How to prove Number of solutions to the Equation $$e^{-x}=x^2-5x+1$$ is $3$ without Graphical approach

My Try:

I considered a Function $$f(x)=(x^2-5x+1)e^x-1$$

I used $$f(-1) =\frac{7-e}{e} \gt 0$$

$$f(-2)=\frac{15-e^2}{e^2}\gt 0$$

But after this essentially we have to use calculator to find $e^n$ value

Can i have any best approach?

Answer 1

With your $f(x)$, $f'(x)=(x^2-3x-4)e^x$ which has zeros exactly at $4,-1$. Look at the values $\lim_{x \to -\infty} f(x),f(-1),f(4),\lim_{x \to \infty} f(x)$. Checking sign changes (which can be done without a calculator) will allow you to apply the intermediate value theorem to get a lower bound on the number of roots. On the other hand Rolle's theorem will provide an upper bound on the number of roots, and the two will turn out to be the same.