Use the Intermediate Value Theorem to show that f(x)=x^4+10x^3-3x^2-5x+4 has a zero in the interval [-1,0]. Explain your reasoning.

Answer 1

Hi Jillian,

First, we generally want to know what the Intermediate Value Theorem says. Textbooks always have very long definitions that can be difficult to follow, so here is a simpler version:

If a function is continuous (no asymptotes or holes or gaps) on an interval from x = a to x = b, then the function must pass through every y-value between the y-value at x = a and the y-value at x = b.

So if f(a) = 7 and f(b) = 10, the function f would hit every y-value between 7 and 10 on the interval.

In your particular problem, the interval is from x = -1 to x = 0. So the first thing we need to do is find the y-values at those x-coordinates:

f(-1) = (-1)⁴+10(-1)³-3(-1)²-5(-1)+4

f(-1) = -3

f(0) = (0)⁴+10(0)³-3(0)²-5(0)+3

f(0)=4

So the function passes through (-1,-3) and (0,4). If you look at those y-values of -3 and 4, and apply the Intermediate Value Theorem (as this function is continuous on the interval; in fact, all polynomial functions are continuous for all real numbers), the theorem would say that the function has to hit every y-value between -3 and 4. This, of course, includes the number 0, which is why you can say the function must have a zero somewhere between x = -1 and x = 0.

I hope this helps!