Question

What is the maximum value of 4(x + 7)(2 - x), over all real numbers x?

Answer 1

The maximum value of function is 81 .

Explanation:

Given function as :

y = 4 (x + 7) (2 - x)

Now, The function can be written as

y = 4 (2 x - x² + 14 - 7 x)

y = 4 ( - x² - 5 x + 14)

y = - 4 x² - 20 x + 56

Now, For maximum value of function , differentiation of y with respect to x

`(\partial y)/(\partial x)` = 0

Or,
`(\partial ( - 4x^(2) - 20 x +56))/(\partial x)` = 0

Or, - 8 x - 20 = 0

Or, - 8 x = 20

∴ x =
`(-20)/(8)`

i.e x =
`(-5)/(2)`

Now, Putting the value of x in the given equation

y = - 4 (
`(-5)/(2)`)² - 20 (
`(-5)/(2)`) + 56

Or, y = - 4 (
`(25)/(4)`) + 50 + 56

Or, y = - 25 + 50 + 56

∴ y = 81

So, The maximum value of function is 81

Hence,The maximum value of function is 81 . Answer