showing a function has exactly one root using Numerical Analysis Methods

Question

Im trying out the Banach Caccioppoli Contradiction Principle but having a few problems..

f(x) = exp(x/2) - 25x^2 How would i show that this function f has exactly one root x^ in (-Infinity,0) ?

Some explanation would be much appeciated!

Many thanks

Answer 1

Let $f(x)=e^{\frac{x}{2}}-25x^2 \,.$

Then $f(0)=1, f(-1)<0$ and $f$ is continuous. IVT tells us there is at least a root.

Now, $f'(x)>0$ on $(0,\infty)$ which means that the function is strictly increasing. Thus it cannot have more than one root...