verify that the function satisfies the three hypotheses of rolle's theoreom on the given interval then find all numbers c that satisfy the conclusion of rolle's theorem.
1) 5-12x+3x^2 [1,3]
To see if Rolle's theorem applies, check
1. if the function is continuous on the closed interval [a, b].
2. if it is differentiable
on the open interval (a,b).
Furthermore,
3. check if f(a)=f(b).
If all the above conditions are satisfied, then there exists some value c between a and be such that f(c)=0.
To verify that the function f(x) = 5 - 12x + 3x^2 satisfies the three hypotheses of Rolle's Theorem on the interval [1, 3], we need to check the following:
1. Continuity: First, we need to make sure that the function is continuous on the closed interval [1, 3].
To do this, we can check if the function is defined and has no breaks or gaps in the interval [1, 3]. In the given function f(x) = 5 - 12x + 3x^2, we observe that it is a polynomial function and therefore defined for all real numbers.
2. Differentiability: Next, we need to check if the function is differentiable on the open interval (1, 3).
To do this, we can differentiate the function and check if it is differentiable on the open interval (1, 3). Taking the derivative of f(x) = 5 - 12x + 3x^2, we get f'(x) = -12 + 6x.
We can see that f'(x) is also a polynomial function and therefore defined for all real numbers, including the open interval (1, 3).
3. Endpoints: Finally, we need to check if the function values at the endpoints of the interval [1, 3] are equal.
To do this, we need to evaluate f(1) and f(3) and check if f(1) = f(3). Evaluating f(1) and f(3) in the given function, f(1) = 5 - 12(1) + 3(1)^2 = -4 and f(3) = 5 - 12(3) + 3(3)^2 = 2.
Since f(1) ≠ f(3), the function does not satisfy the endpoint condition of Rolle's Theorem.
Therefore, the function does not satisfy all the hypotheses of Rolle's Theorem on the interval [1, 3].
As a consequence, we cannot find any number c that satisfies the conclusion of Rolle's Theorem, as the theorem itself is not applicable in this case.