Finding intervals containing solutions to a equation, without using a graph How can I find all intervals containing solutions to the equation $4x^2-e^x=0$, without using a graph? I know how to solve this problem, but not without a graph. Thanks.
HINT: 

Find the derivative of the function $f(x) = 4x^2 - e^x$ and set it to $0$ to find it's local minimum or maximum.  You can solve the new equation using Lambert W function. 

Then use the second derivative test to check if the point is maxima or minima. You should be able to obtain 1 local minima and 1 local maxima. First we have a minima, so that means that the function is strictrly decresing in that interval, now plug the minima into your equation to find if the value of the function is smaller than 0. Yes, it is and that means that there's one zero in that interval. Then check the interval between the minima and the maxima, if they are on different sides of the x-axis, the we have another zero. And at last check if the maxima is bigger than $0$, because in the interval $(maxima, \infty)$, the function is decreasing. So if for the maxima, function has value bigger than $0$, we have another zero in that interval.

So in the end there are 3 zeros of the function. And we have one in each of the intervals:
$(-\infty,minima), (minima,maxima), (maxima,\infty)$
If you plug in values randomly for $f(x)$ and you're able to get $f(a).f(b)&lt;0$ then $(a,b)$ is the interval in which roots of the equation would lie.