How do you determine if rolles theorem can be applied to #f(x) = 2x^2 − 5x + 1# on the interval [0,2] and if so how do you find all the values of c in the interval for which f'(c)=0?

Question

Evelyn Agard · Accepted Answer

The Rolles theorem says that if: then at least one #cin(a,b)# as if #f'(c)=0# exists. So:

Elijah Abram · Answer

undefined To determine if Rolle's Theorem can be applied to $ f(x) = 2x^2 - 5x + 1 $ on the interval $[0,2]$, check if the function satisfies the conditions of Rolle's Theorem. The function must be continuous on the closed interval $[0,2]$ and differentiable on the open interval $(0,2)$, and $ f(0) = f(2) $.

Calculate $ f'(x) $, the derivative of $ f(x) $, and find the critical points in the interval $(0,2)$ by setting $ f'(x) = 0 $ and solving for $ x $. Then, evaluate the critical points within the interval $[0,2]$ to find the values of $ c $ for which $ f'(c) = 0 $.

First, find the derivative:
$$ f'(x) = 4x - 5 $$

Setting $ f'(x) $ equal to zero:
$$ 4x - 5 = 0 $$
$$ 4x = 5 $$
$$ x = \frac{5}{4} $$

This critical point falls within the interval $[0,2]$, so Rolle's Theorem applies.

Therefore, $ c = \frac{5}{4} $ is the only value in the interval $[0,2]$ for which $ f'(c) = 0 $.