usethe intermediate theorem to show that the polynomial function has a zero in the given interval
f(x)=18x^4-8x^2+9x-1;[0,3)
can you please me how you got the answer
f(0) = -1
f(3) > 0
so, to get from -1 to a positive value, it must pass through zero, because f is continuous.
To apply the Intermediate Value Theorem, we need to show that the polynomial function f(x) has different signs at the endpoints of the given interval [0, 3) and that it is a continuous function.
1. Evaluating f(0):
f(0) = 18(0)^4 - 8(0)^2 + 9(0) - 1 = -1
2. Evaluating f(3):
f(3) = 18(3)^4 - 8(3)^2 + 9(3) - 1 = 194
Now, we have f(0) = -1 and f(3) = 194. Since these values have different signs, we know that the polynomial function f(x) must have at least one zero (i.e., root) in the interval [0, 3).
Furthermore, to show continuity, we can observe that f(x) is a polynomial and, therefore, continuous over its entire domain.
So, based on the fact that f(x) changes sign between f(0) and f(3), we can confirm that the polynomial function f(x) has a zero (root) in the given interval [0, 3).