Use the intermediate value theorem to determine whether the polynomial function has a zero in the given interval.
f(x)=8x^5-4x^3-9x^2-9;[1,2]
Calculate f(x) at x = 1 and 2, and see if it changes sign between those two points.
To use the intermediate value theorem to determine whether the polynomial function has a zero in the given interval, we need to evaluate the function at the endpoints of the interval and check if the function changes sign.
First, let's evaluate the function at the lower endpoint of the interval, which is x = 1:
f(1) = 8(1)^5 - 4(1)^3 - 9(1)^2 - 9
= 8 - 4 - 9 - 9
= -14
Next, let's evaluate the function at the upper endpoint of the interval, which is x = 2:
f(2) = 8(2)^5 - 4(2)^3 - 9(2)^2 - 9
= 8(32) - 4(8) - 9(4) - 9
= 256 - 32 - 36 - 9
= 179
Now, we need to check if the function changes sign between these two values (-14 and 179). Since the function evaluated at x = 1 is negative and the function evaluated at x = 2 is positive, we can conclude that the function crosses the x-axis at some point between 1 and 2.
Therefore, according to the intermediate value theorem, the polynomial function f(x) = 8x^5 - 4x^3 - 9x^2 - 9 has a zero in the given interval [1, 2].