Question

Use the intermediate value theorem to find the value of c such that f(c) = M.
f(x) = x^2 - x + 1 text( on ) [-1,12]; M = 21

Answer 1

First, check to be sure that M is in the range of f over the interval:

f(-1) = 3
f(12) = 133
so 21 is in the range.

Since f is continuous, and changes from 3 to 133, it must attain the value of 21 somewhere in there. That is, there is some c such that

f(x) = 21
To find c, just solve the equation

x^2-x+1 = 21
x^2-x-20 = 0
(x-5)(x+4) = 0
x = -4,5

SO, f(5) = 21 and 5 is in [-1,12]

f(-4) = 21 too, but it's outside the interval of interest.