f(x)=9x^3+2x^2-4x+7; [-2,-1]
Value of f(-2)
f(-2)=
Value of f(-1)
f(-1)=
Does f have a zero in the given interval?
Value of f(-2)
f(-2)=
Value of f(-1)
f(-1)=
Does f have a zero in the given interval?
-
One of the applications of the intermediate value theorem allows you to determine if a polynomial has a zero on a given interval. Basically, if a function is continuous and at the beginning of the interval it's positive and at the end of the interval it's negative (or vice versa), then the function must have had a zero somewhere on the interval. As a stand alone theorem, it doesn't tell you where the zero is, or even how many there are, but it tells you that there is at least one zero on the interval.
f(-2) = -72 + 8 + 8 + 7 = -49
f(-1) = -9 + 2 + 8 + 7 = 8
The curve goes from (-2, - 49) to (-1, 8), so at some point on the interval [-2, -1], there had to be a zero. As a stand alone theorem, it doesn't tell you where the zero is, or even how many there are, but it tells you that there is at least one zero on the interval.
f(-2) = -72 + 8 + 8 + 7 = -49
f(-1) = -9 + 2 + 8 + 7 = 8
The curve goes from (-2, - 49) to (-1, 8), so at some point on the interval [-2, -1], there had to be a zero. As a stand alone theorem, it doesn't tell you where the zero is, or even how many there are, but it tells you that there is at least one zero on the interval.