Want to make sure I am doing this correctly.
Use the Intermediate Value Theorem for polynomials to show that the polynomial function f(x)has zero between the numbers given.
f(x)=3x^2-x-4 ; 1 and 2
3(1)^2-1-4= - 2 below zero
3(2)^2-1-4= 6 above zero
Am I doing this problem correct
correct. It can't get from -2 to 6 without going through 0 somewhere in between.
You also need to mention that f(x) is continuous. That is necessary.
Thank you for your help.
Yes, you are on the right track and your approach to using the Intermediate Value Theorem for polynomials is correct.
To use the Intermediate Value Theorem, you need to evaluate the function at the two given numbers, in this case, 1 and 2, and determine whether the function changes sign between those two values. If it does, then the theorem guarantees the existence of at least one zero (root) between those two numbers.
Let's go through it step by step using the function f(x) = 3x^2 - x - 4:
1. Evaluate the function at x = 1:
f(1) = 3(1)^2 - 1 - 4
= 3(1) - 1 - 4
= 3 - 1 - 4
= -2
So, f(1) = -2, which is below zero.
2. Evaluate the function at x = 2:
f(2) = 3(2)^2 - 2 - 4
= 3(4) - 2 - 4
= 12 - 2 - 4
= 6
So, f(2) = 6, which is above zero.
Since the function changes sign between f(1) and f(2) (from below zero to above zero), we can conclude that there must exist at least one zero of the function f(x) between x = 1 and x = 2.
Therefore, you have correctly determined that the polynomial function has a zero between the numbers 1 and 2. Well done!