Consider the function f(x) = -2x3 + x2 – 3x + 4 on the interval [-2,4] and that f(c) = 10

Question

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem

Answer 1

The intermediate value theorem states that if f is continuous on [a,b], and d is number such that d is between f(a) and f(b), then there must be some number, c, in the interval [a,b], such that f(c)=d. In other words, since f is continuous on [a,b], and you know the endpoints give you the output values f(a) and f(b), every number between these outputs must have at least one corresponding input - the function can't "skip" past an output.

In this case, [a,b] = [-2,4].

f is a polynomial, which are continuous on the entire real line, and so certainly continuous on [-2,4]

f(-2) = 30, and f(4) = -120, and so for every value between -120 and 30, we know there is at least one input in [-2,4] that will yield that output value.

10 is certainly between -120 and 30, and so we know there is at least one input on [-2,4] whose output is 10.

To find it, we solve the equation

-2x^3+x^2-3x+4 = 10, or

-2x^3+x^2-3x-6=0.

This is a cubic equation - although a "cubic equation", similar to the quadratic equation, exists, it's so "messy" it's rarely used.

If you remember pre-calculus, there is a 'rational root' theorem that gives a list of values you can check, assuming a rational root exists.

Or, you can graph the function -2x^3+x^2-3x-6 and find its zeros - this is the quickest approach.

I found that -2(-1)^3+(-1)^2-3(-1)-6 = 2+1+3-6=0, and so x=-1 is a solution,

and so f(-1) = 10.