Question

The function `f(x) = x^3 - 6x^2 + 9x + 25` has

Options

• a maxima at x = 1 and a minima at x = 3

• a maxima at x = 3 and a minima at x = 1

• no maxima, but a minima at x = 1

• a maxima at x = 1, but no minima

Answer 1

a maxima at x = 1 and a minima at x = 3

Explanation:

`f(x) = x^3 - 6x^2 + 9x + 25`

`f^'(x) = 3x^2 - 12x + 9`

`f^('')(x) = 6x - 12`

To get the stationary points ⇒ `f^'(x)` = 0

∴ `3x^2 - 12x + 9` = 0

`x^2 - 4x + 3` = 0

`(x - 1)(x - 3)` = 0

`x = 1` or `x = 3`

To get the maxima and minima of a function,

`f^('')(x) = 6x - 12`

At `x = 1, f^('')(1) = 6(1) - 12 = 6 - 12 = - 6 < 0`

∴ Maxima of `f(x)` exists at `x` = 10.

At `x = 3, f^('')(3) = 6(3) - 12 = 18 - 12 = 6 > 0`.

∴ Minima of `f(x)` exists at `x` = 3.