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
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.
Well, according to the Intermediate Value Theorem, if a function f(x) is continuous on a closed interval [a, b], and M is any number between f(a) and f(b), then there exists at least one value c in the interval [a, b] such that f(c) = M.
In this case, f(x) = x^2 - x + 1 on the interval [-1, 12], and M = 21. Let's check if M falls between f(-1) and f(12):
f(-1) = (-1)^2 - (-1) + 1 = 1 + 1 + 1 = 3
f(12) = (12)^2 - 12 + 1 = 144 - 12 + 1 = 133
So, since 21 is between 3 and 133, there must exist at least one value c in the interval [-1, 12] such that f(c) = 21. However, determining the exact value of c would require more information, such as additional conditions or solving the equation f(c) = M.
To use the Intermediate Value Theorem to find the value of c such that f(c) = M, we need to check if the function f(x) = x^2 - x + 1 takes on the value M = 21 between the interval [-1, 12].
First, let's evaluate the function f(x) at the endpoints of the interval:
f(-1) = (-1)^2 - (-1) + 1 = 1
f(12) = (12)^2 - 12 + 1 = 133
Since M = 21 lies between f(-1) and f(12), the Intermediate Value Theorem guarantees that there exists at least one value c in the interval (-1, 12) such that f(c) = M = 21.
To use the intermediate value theorem to find the value of c such that f(c) = M, we need to follow these steps:
1. Verify that f(x) is continuous on the interval [-1, 12]. In this case, f(x) = x^2 - x + 1, which is a polynomial function and therefore continuous everywhere. So f(x) is continuous on the interval [-1, 12].
2. Determine the value of f(-1) and f(12). Calculate f(-1) by substituting x = -1 into the function f(x): f(-1) = (-1)^2 - (-1) + 1 = 1 + 1 + 1 = 3. Calculate f(12) by substituting x = 12 into the function f(x): f(12) = (12)^2 - 12 + 1 = 144 - 12 + 1 = 133.
3. Check if the value M = 21 lies between f(-1) and f(12). In this case, 3 < 21 < 133, so M = 21 lies between f(-1) and f(12).
4. Apply the intermediate value theorem. Since f(x) is continuous on the interval [-1, 12], and M = 21 lies between f(-1) and f(12), there must exist a value c in the interval [-1, 12] such that f(c) = M.
Therefore, the intermediate value theorem guarantees the existence of a value c in the interval [-1, 12] such that f(c) = 21. However, to find the exact value of c, we would need to use numerical methods such as bisection method or Newton's method.