# Roots of an equation (TIFR 2013 problem 3)

Problem

The equation $$x^3+3x-4=0$$ has exactly one real root.

Discussion

Hint: Try differentiating. Is the function monotonic?

Note that cubic equation has at least one real root (fundamental theorem of algebra provides for three roots counting repetitions, real or complex; for equations with real coefficients, the complex roots appear in conjugate pairs. Hence a cubic equation with real coefficients must have at least one real root).

Now differentiating $$P(x) = x^3 + 3x – 4$$ we have the following: $$\frac {d} {dx} P(x) = 3x^2 + 3$$