The equation is .

Consider the function .

Differentiate on each side with respect to .

Find the critical points.

Equate .

Critical point of  is

The function is a decreasing function, when .

The function is an increasing function, when .

If , then  for all values of , and hence it has no real roots.

If , then  has a single real zero at .

If , then  for all values of , and hence it has no real roots.

Find -values to the left and right of , where .

Use Intermediate Value Theorem to infer that  has two real roots.

Consider .

At , .

Consider .

At , .

Since , apply the intermediate theorem to state that there must be some  in such that .

Observe the above cases find, that the function never have more than two real roots.

Thus, the function  has at most two real roots.

The function  has at most two real roots.