How can we find the number of solutions of the polynomial $f(x)= x^3 - 3x +b$ that lie in the interval $[-1,1]$?

Question

How many solutions of the polynomial $f(x)= x^3 - 3x +b$ lie in the interval $[-1,1]$?

a) None

b) At most one

c) One

d) At least one

Answer 1

HINT: $$f'(x)=3x^2-3$$ $$f''(x)=6x$$ and solve the equation $$f'(x)=0$$

Answer 2

Without doing math, we can play everyone's favorite game, "Beat the Multiple Choice Test."

If $b=0$, then $x=0$ is a root. This eliminates a). If $b=$ a billion then, by imagining the graph, we see that there are no roots in the interval, so c) and d) are eliminated. That leaves b).

Answer 3

Let's examine $f'(x) = 3x^2 - 3$. This has roots at $x=-1$ and $x=1$, and $f(x)$ is strictly decreasing on the interval $[-1,1]$, so it can have at most one root in this interval (if it had $2$ or more, we would have more critical points on the interior of this interval). This eliminates answer choice (d).

We cannot say if it necessarily has a zero in this interval, however, as $f(-1)=2+b$ and $f(1)=-2+b$ are both positive (resp. negative) for $b>2$ (resp. $b<2$). This eliminates answer choices (a) and (c), leaving us with (b) At most one as the correct answer.