I am preparing for an entrance and this question has taken too much of my time.
Suppose function $f : \mathbb R \to \mathbb R$ is given by $$f(x) = x^3 - 3x + b$$ Find the number of points in the closed interval $[-1,1]$ at which $f(x)=0$.
How to find the number of roots of $f$? I have found the roots of $f'$ are $\pm 1$. Its a MCQ and answer says atmost one but mine is coming to be 2. I wanted to see how derivative can help in finding roots of a polynomial
Hint:
As $f'(x)=3(x^2-1)<0$ on $(-1,1)$, $f(x)$ is decreasing on $[-1,1]$, hence it has at most one root on this interval.
It has exactly one root if and only if $f(1)$ and $f(-1)$ have opposite signs. This will depend on the value of $b$.