Question

What are the extrema of `f(x)=(x - 4)(x - 5)` on `[4,5]`?

Answer 1

The extremum of the function is (4.5 , -0.25)

Explanation:

`f(x) = (x-4)(x-5)` can be rewritten to `f(x) = x^2 - 5x - 4x + 20 = x^2-9x+20`.

If you derivate the function, you will end up with this:
`f'(x) = 2x - 9`.
If you don't how to derivate functions like these, check the description further down.

You want to know where `f'(x) = 0` , because that's where the gradient = 0.

Put `f'(x) = 0` ;
`2x - 9 = 0`
`2x = 9`
`x = 4.5`

Then put this value of x into the original function.
`f(4.5) = (4.5 - 4)(4.5-5)`
`f(4.5) = 0.5 * (-0.5)`
`f(4.5) = -0.25`

Crach course on how to derivate these types of functions:
Multiply the exponent with the base number, and decrease the exponent by 1.

Example:
`f(x) = 3x^3 - 2x^2 - 2x + 3`
`f'(x) = 3 * 3x^(3-1) - 2 * 2x^(2-1) - 1 * 2x^(1-1)`
`f'(x) = 9x^2 - 2x - 2x^0`
`f'(x) = 9x^2 - 2x - 2`