Use the intermediate value theorem to show that the polynomial function has a zero in the given interval.
f(x)=4x^3+6x^2-7x+1; [-4,-2]
f(-4)=
f(-4) = -131
f(-2) = 7
IVT says that f(x) assumes every value between -131 and 7 on [-4,-2]. In particular, there is some c in [-4,-2] where f(x) = 0.
To use the intermediate value theorem to show that the polynomial function has a zero in the given interval, we need to find the values of f(x) at the endpoints of the interval and check if they have opposite signs.
First, let's find f(-4):
f(x) = 4x^3 + 6x^2 - 7x + 1
Substituting x = -4 into the function:
f(-4) = 4(-4)^3 + 6(-4)^2 - 7(-4) + 1
= 4(-64) + 6(16) + 28 + 1
= -256 + 96 + 28 + 1
= -256 + 96 + 29
= -131
Therefore, f(-4) = -131.
Now, let's check the sign of f(-4) and f(-2) to see if they have opposite signs.
f(-4) = -131 --> negative value
f(-2) = f(-2) = 4(-2)^3 + 6(-2)^2 - 7(-2) + 1
= 4(-8) + 6(4) + 14 + 1
= -32 + 24 + 14 + 1
= 7
Since f(-4) is negative and f(-2) is positive, we have opposite signs.
According to the intermediate value theorem, since f(x) is continuous on the interval [-4, -2], and f(-4) is negative while f(-2) is positive, there must exist at least one value of x between -4 and -2 for which f(x) is equal to zero.
To use the intermediate value theorem to show that the polynomial function has a zero in the given interval, we need to find the values of f(x) at the endpoints of the interval and check if one of them is positive while the other is negative (or vice versa). If this condition is met, then by the intermediate value theorem, since the function is continuous, there exists at least one value c in the interval [-4, -2] such that f(c) = 0.
Let's calculate f(-4) to check its value:
f(-4) = 4(-4)^3 + 6(-4)^2 - 7(-4) + 1
f(-4) = 4(-64) + 6(16) + 28 + 1
f(-4) = -256 + 96 + 28 + 1
f(-4) = -256 + 96 + 28 + 1
f(-4) = -131
Since f(-4) = -131, which is negative, we can see that f(x) changes sign between -4 and some value in the interval [-4,-2]. Therefore, by the intermediate value theorem, we can conclude that the polynomial function f(x) = 4x^3 + 6x^2 - 7x + 1 has a zero in the given interval [-4,-2].