In exercises 83–86, verify that the intermediate value theorem applies to the indicated interval and find the value of guaranteed by the theorem.

The intermediate value theorem states that suppose f is a function continuous on every point of the interval [a, b], for any L between the values f(a) and f(b), there exist at least one number, c, in [a, b] for which f(c) = L.

83.
Given

on the interval [0, 5], with f(c) = 11


and

Clearly, f(c) = 11 is between f(0) = -1 and f(5) = 29

Thus, the intermediate value theorem is satisfied.

To find the value of c, we solve the equation

as follows:


Because, c is in the interval [0, 5], therefore, the value of c is 3.

84.
Given

on the interval [0, 3], with f(c) = 0


and

Clearly, f(c) = 0 is between f(0) = 8 and f(3) = -1

Thus, the intermediate value theorem is satisfied.

To find the value of c, we solve the equation

as follows:


Because, c is in the interval [0, 3], therefore, the value of c is 2.

85.
Given

on the interval [0, 3], with f(c) = 4


and

Clearly, f(c) = 4 is between f(0) = -2 and f(3) = 19

Thus, the intermediate value theorem is satisfied.

To find the value of c, we solve the equation

as follows:


Therefore, the value of c is 2.

86.
Given

on the interval , with f(c) = 6


and

Clearly, f(c) = 6 is between and

Thus, the intermediate value theorem is satisfied.

To find the value of c, we solve the equation

as follows:


Because, c is in the interval , therefore, the value of c is 3.