Question

Using Rolle's theorem for the following function, find all values c in the given int If there are multiple values, separate them using a comma. f(x)=2x³+15x²-36x-1

Answer 1

To apply Rolle's theorem, we ensure the function meets the theorem's conditions, find its derivative, and then solve for c in f'(c) = 0. The solutions represent the values where the function's slope is zero, provided they fall within the given interval.

Step-by-step explanation:

To apply Rolle's theorem to the function f(x) = 2x³ + 15x² - 36x - 1, we must first ensure that the function fulfills the theorem's conditions:

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• The function f(x) is continuous on the closed interval [a, b].
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• The function f(x) is differentiable on the open interval (a, b).
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• f(a) = f(b) for some points a and b.

Assuming these conditions are satisfied, we can find the derivative of the function, f'(x) = 6x² + 30x - 36. Rolle's theorem states that there is at least one c in the interval (a, b) such that f'(c) = 0.

Now, we need to solve the equation f'(c) = 6c² + 30c - 36 = 0. This is a quadratic equation and can be solved by factoring, completing the square, or using the quadratic formula. After finding the solutions, we ensure that they belong to the interval (a, b) and are therefore valid values that satisfy Rolle's theorem.

After solving, we might find multiple values of c that satisfy f'(c) = 0. If there are multiple values, we will list them separated by commas.