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Which theorem can be used to determine whether a function f(x) has any zeros in a given interval?

O A. Rolle's Theorem

OB.

O C.

O D. Mean value theorem

< Question 12 of 20 >

Intermediate value theorem

Extreme value theorem

To apply this theorem, evaluate the function f(x) = x++ 8x + 2 at each endpoint of the interval [-1, 0].

f(-1) = (Simplify your answer.)

f(0) = (Simplify your answer.)

According to the intermediate value theorem, f(x) = x4 + 8x + 2 has

Voc

in the given interval.

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Now, determine whether there can be more than one zero in the given interval.

Rolle's Theorem states that for a function f(x) that is continuous at every point over the closed interval [a,b] and differentiable at every point of its interior (a,b), if f(a) = f(b), then there is at least one number c in

(a,b) at which f'(c) = 0.

Find the derivative of f(x)= x + 8x + 2.

f'(x) =

Can the derivative of f(x) be zero in the interval [-1, 0]?

Fig: 1