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Show that the function f(x)= x² + 8x + 2 has exactly one zero in 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

O

Question 12 of 20 >

Yes

No

in the given interval.

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)=x² + 8x + 2.

f'(x) =

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

O No

O Yes

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This question: 1 point(s) possible

The function f(x) = x4 + 8x + 2 has at least one zero at some point x = a in the interval [-1, 0]. According to Rolle's Theorem, can there be another point x = b in this interval where f(a) = f(b) = 0?

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Thus, since the intermediate value theorem shows that f(x) = x++ 8x + 2 has at least one zero in the interval [-1, 0] and Rolle's Theorem shows that there cannot be two points x = a and x = b for which

f(a) = f(b) in this interval, the function f(x) has exactly one zero in the interval [-1, 0].