find the intervals in which the following functions is strictly increasing or strictly decreasing F(x)=4x3-6x2-72x+30
Answers
Answer: Strictly increasing interval: (-inf, -2) and (3, inf)
Strictly decreasing interval: (-2, 3)
Step-by-step explanation:
To determine intervals where the function strictly increases or decreases, we need to first find the derivative of the function.
F(x)= 4x^3-6x^2-72x+30
Differentiating with respect to x, we get:
F'(x)= 12x^2 - 12x - 72
Factorizing the above derivative and equating to zero, we get:
12(x+2)(x-3)=0
x= -2, 3
These are the critical points. These points divide the domain into 3 intervals where the function strictly increases or decreases.
So, we get three intervals as follow:
(-inf, -2), (-2, 3), and (3, inf)
Now, we need to check which interval is strictly increasing and which is strictly decreasing.
if x1 < x2 => F(x1) < F(x2) then function is strictly increasing in that interval
and if x1 < x2 => F(x1) > F(x2) the function is decreasing.
For interval (-2, 3)
Suppose, x1= 0 and x2= 1
F(x1=0)= 4*0^3 - 6*0^2 - 72*0 +30 =30
F(x2=1)= 4*1^3 - 6* 1^2 - 72*1 +30 = 4 - 6 - 72 + 30 = -44
Since, x1 < x2 but F(x1) > F(x2) , therefore, the function is strictly decreasing in this interval.
Similarly, we can check for other intervals.