Question

How do you determine whether the function `f(x)= 5+12x- x^3` is concave up or concave down and its intervals?

Answer 1

Refer the explanation section

Explanation:

Given -
y = `- x^3 + 12x + 5`

Find the first 2 derivatives -

`dy/dx` = `- 3x^2 + 12`
`(d^2y)/dx^2` = `- 6x`

Set the 1st derivative to zero to find for what value of 'x' the curve turns.

`dy/dx =>` `- 3x^2 + 12`=0
`-3x^2` = - 12
`x^2` = `(-12)/-3`= 4
x = `+-sqrt 4`

x = 2
x = - 2

At x = 2 and x = - 2 the curve turns. To determine whether it turns upwards or downwards, substitute the values in the 2nd derivative.

At x = 2 ; `(d^2y)/dx^2` = `- 6x` =`- 6 xx 2`= - 12 < 0

The curve has a maximum at x = 2. In the immediate proximity the curve is concave downwards.

At x = - 2 ; `(d^2y)/dx^2` = `- 6x` =`- 6 xx - 2`= 12 > 0

The curve has a Minimum at x = - 2. In the immediate proximity the curve is concave upwards.

Point of inflection separates concavity from convexity. To the Point of inflection, set the 2nd derivative equal to zero.

`(d^2y)/dx^2 = 0 => - 6x = 0`
x = 0

At x = 0 , there is point of inflection.

-`oo` < x < 0 ; curve is concave upwards
`oo` > x > 0 ; curve is concave downwards.

graph{-x^3 + 12 x + 5 [-74.04, 74.1, -37.03, 37]}