Question

Use the intermediate value theorem to find the value of c such that f(c) = M. f(x) = x^2 - x + 1 text( on ) [1,8]; M = 21 c =

Answer 1

`c = 5`

Explanation:

Given

`f(c) = M`

`f(x) = x^2 - x + 1`

`Interval: [1,8]`

`M = 21`

Required

Find c using Intermediate Value theorem

First, check if the value of M is within the given range:

`f(x) = x^2 - x + 1`

`f(1) = 1^2 - 1 + 1`

`f(1) = 1`

`f(x) = 8^2 - 8 + 1`

`f(x) = 57`

`1 \le M \le 57`

`1 \le 21 \le 57`

M is within range.

Solving further:

We have:

`f(c) = f(x) = M`

`f(x) = 21`

Substitute 21 for f(x) in
`f(x) = x^2 - x + 1`

`21 = x^2 - x + 1`

Express as quadratic function

`x^2 - x + 1 - 21 = 0`

`x^2 - x - 20 = 0`

Expand

`x^2 + 4x - 5x - 20`

`x(x+4)-5(x+4)=0`

`(x - 5)(x+4) = 0`

`x - 5 = 0` or
`x + 4= 0`

`x = 5` or
`x = -4`

The value of
`x = -4` is outside the
`Interval: [1,8]`

So:

`x = 5`

`f(c) = f(x) = M`

`f(c) = f(5) = 21`

By comparison:

`c = 5`