Question thirteen

Question

Use the intermediate value theorem to show there is a zero of f(x)=x^2 - 8 between x=2 and x=3

Answer 1

The intermediate value theorem states that you must have every intermediate value of a function on an interval between two endpoints so long as the function is continuous. So, to prove there is a zero of f(x)=x²-8 between x=2 and x=3, simply take the values of the function at x=2 and x=3 and show that the function values surround zero. This will prove there is a zero between the two x values since we know the function x²-8 is continuous:

f(2)=(2)²-8= -4 < 0 (less than 0)

f(3)=(3)²-8=1 > 0 (greater than 0)

Therefore there must be a zero on the interval 2≤x≤3