How do you determine whether the function #f(x)= 5+12x x^3# is concave up or concave down and its intervals?
1 Answer
Refer the explanation section
Explanation:
Given 
y =
Find the first 2 derivatives 
Set the 1st derivative to zero to find for what value of 'x' the curve turns.
x =
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 ;
The curve has a maximum at x = 2. In the immediate proximity the curve is concave downwards.
At x =  2 ;
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.
x = 0
At x = 0 , there is point of inflection.

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