# 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 #mathbf(x<1)#:**

#f''(0)=-24# Since this is

#<0# , the function is concave on the interval#(-oo,1)# .

**When #mathbf(x>1)#:**

#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