Question

Show that the equation x^4 + 4x + c = 0 has at most two real roots.

Answer 1

Step 1 :

The equation is .

Consider .

This is the critical point of .

The function is a decreasing function, when , and

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 .

Find - values to the left and right of , where

Step 2 :

Use intermediate value theorem to inform 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, notice that, the function never have more than two real roots.

Thus, the function has at most two real roots.

Solution :

The equation has at most two real roots.