Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem.

f (x) = x^2 + x - l, [0, 5], f (c) = 11

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Aurora

Mathematics

29 September, 13:03

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem.

f (x) = x^2 + x - l, [0, 5], f (c) = 11

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Answers (1)

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Francesca Stanton · Answer 1

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Step-by-step explanation:

Remember that the Intermediate Value Theorem (IVT) states that if f is a continuous function on [a, b] and f (a)
Now, to apply the theorem, we have that f (0) = 0²+0-1=-1, f (5) = 5²+5-1=29, then f (0) = -1<11<29=f (5). Additionally, f is continous since it is a polynomial. Then the IVT applies, and such c exists.

To find, c, solve the quadratic equation f (c) = 11. This equation is c²+c-1=11. Rearranging, c²+c-12=0. Factor the expression to get (c+4) (c-3) = 0. Then c=-4 or c=3. - 4 is not in the interval, then we take c=3. Indeed, f (3) = 3²+3-1=9+3-1=11.

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem.f (x) = x^2 + x - l, [0, 5], f (c) = 11

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem.

f (x) = x^2 + x - l, [0, 5], f (c) = 11