Question 12 of 20 >

Show that the function f(x)=x² + 5x+3 has exactly one zero in the interval [-1, 0].

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

O A. Mean value theorem

O B.

Rolle's Theorem

O C.

Extreme value theorem

OD. Intermediate value theorem

To apply this theorem, evaluate the function f(x)=x² + 5x+3 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)=x² + 5x + 3 has

...

in the given interval.

This quiz: 20 point(s) possible

This question: 1 point(s) possible

Comment

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 + 5x + 3.

f'(x) =

Can the derivative of f(x) be zero in the interval [-1, 0]?/n< Question 12 of 20 >

Show that the function f(x)=x² + 5x +3 has exactly one zero in the interval [-1, 0].

t(-1)= || (Simplity your answer.)

f(0) = (Simplify your answer.)

According to the intermediate value theorem, f(x)=x² + 5x + 3 has

Yes

No

in the given interval.

This quiz: 20 point(s) possible

This question: 1 point(s) possible

Added

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 + 5x + 3.

f'(x) =

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

The function f(x)=x² + 5x+3 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?

O No

Yes

Comment

Thus, since the intermediate value theorem shows that f(x)=x² + 5x + 3 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].