Derivative Application

Question

f(x)=x^3+6x^2-36x+8

f'(x)=3x^2+12x-36

f''(x)=6x+12

 

By using interval notation, On what interval is f(x) increasing including endpoints in interval? Downward and upward concavity? 

Please help me, I dont know how to solve this using an interval notation 

Answer 1

INCREASING/DECREASING:

 

The rule is: Increasing/Decreasing Test:

 

(a) If f' x > 0 on an interval, then f is increasing on that interval.

(b) If f' x < 0 on an interval, then f is decreasing on that interval.

 

In English: The first derivative equation will tell you where  the function is increasing and where it is decreasing. 

 

Steps:

First find the critical points if the first derivative (i.e. when the first derivative equal to 0):

 

3x^2+12x-36 = 0

 

x = -6 and x=2 

 

So we have 3 intervals : (-inf,-6), (-6,2), (2,inf) The number of interval is always one more than the number of critical points. 

 

Now plug the values from this intervals into first derivative to see if its positive or negative:

 

Lets pick first interval x< -6: let't pick -7: 3*(-7)^2+12*-7-36 = 27 (this value is positive). So the function is increasing on interval (-inf, -6) 

 

Doing the same for second interval: (-6,2) Lets pick a point 0 since its easy to calculate: 3*0^2+12*0-36 = -36.

The first derivative is less than 0 so the function is decreasing on the whole interval (-6,2)

 

Doing the same for third interval: (2,inf) Lets pick a point 5 : 3*5^2+12*5-36 = 99
The first derivative is more than 0 so the function is increasing on the whole interval (2,inf)

 

 

CONCAVITY:

 

Concavity Test

 

(a) If f'' (x) > 0 for all x in I,then the graph of f is concave upward on I.

(b) If f'' (x) < 0 for all x in I, then the graph of f is concave downward on I.

 

In English: The second derivative equation tells you the concavity.

 

Steps are same as before, you just use the second derivative:

 

First find where the critical points are (i.e. where the second derivative is equal to 0):

 

6x+12= 0

x = -2

 

so we have 2 intervals (-inf,-2) and (-2,inf):

 

same as before now just plug the points from each interval into the second derivative to see if it's > 0 or < 0:

 

first interval (-inf,-2): Let's pick -5 : 6*-5+12 = -18 so less than 0 and therefor is concave down.

 

second interval (-2,inf): Let's pick 0 since it's easy to calculate: 6*0+12 = 12 so more than 0 and therefor is concave up.