intermediate value theorem

Question

use the intermediate value theorem to determine whether the equation g(x)=2x^3-x^2+2x-3 has a real zero between 0 and 2

Answer 1

f(x) = 2x³ - x² + 2x - 3

 

f(0) = 2·0³ - 0² + 2·0 - 3 = -3

f(2) = 2·2² - 2² + 2·2 - 3 = 5

 

The intermediate value theorem states that the function f(x) on the interval [0, 2] will take on all values between f(0) and f(2) at least once.  Since 0 is between -3 and 5, there must be an x between x=0 and x=2 where f(x) = 0.  So yes, there is at least one real zero between x=0 and x=2.