For what values of x is #f(x)= 4x^3-12x^2 # concave or convex?
1 Answer
Concave on
Explanation:
The convexity and concavity of a function and determined by the sign of the second derivative.
- If
#f''(a)>0# , then#f(x)# is convex at#x=a# . - If
#f''(a)<0# , then#f(x)# is concave at#x=a# .
First, find the second derivative.
#f(x)=4x^3-12x^2#
#f'(x)=12x^2-24x#
#f''(x)=24x-24#
The second derivative could change signs whenever it is equal to
#24x-24=0#
#x=1#
The convexity/concavity could shift only at this point. Thus, from here, we can determine on which intervals the function will be uninterruptedly convex or concave.
Use test points around
When
#f''(0)=-24# Since this is
#<0# , the function is concave on the interval#(-oo,1)# .
When
#f''(2)=24# Since this is
#>0# , the function is convex on the interval#(1,+oo)# .
Always consult a graph of the original function when possible:
graph{4x^3-12x^2 [-2 5, -19.9, 5.77]}
The concavity does seem to shift around the point