For what values of x is #f(x)= 4x^3-12x^2 # concave or convex?

Question

Charles Arocha · Accepted Answer

Concave on #(-oo,1)#; convex on #(1,+oo)#

The convexity and concavity of a function and determined by the sign of the second derivative.

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 #0#. Find that point by setting the second derivative equal to #0#.

#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 #x=1#:

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 #x=1#. When #x<1#, the graph points downward, in the #nn# shape characteristic of concavity. When #x>1#, the graph points upward in the #uu# shape characteristic of convexity.

Ezekiel Alexander · Answer

undefined To determine where the function $ f(x) = 4x^3 - 12x^2 $ is concave or convex, you need to find the second derivative of the function and analyze its sign. The second derivative will help you identify the concavity of the function. First, find the first derivative of the function: $$ f'(x) = 12x^2 - 24x $$ Now, find the second derivative: $$ f''(x) = 24x - 24 $$ To determine the concavity or convexity, examine the sign of the second derivative. If $ f''(x) > 0 $, the function is concave up (convex) on that interval. If $ f''(x) < 0 $, the function is concave down (concave) on that interval. Setting $ f''(x) = 0 $ gives you the inflection points. So, solve for $ x $ in $ f''(x) = 0 $: $$ 24x - 24 = 0 $$ $$ x = 1 $$ Now, test the intervals: 1. Test $ x < 1 $: Choose $ x = 0 $ (which is less than 1) Substitute $ x = 0 $ into $ f''(x) $: $ f''(0) = 24(0) - 24 = -24 $ Since $ f''(0) < 0 $, the function is concave down on $ (-\infty, 1) $. 2. Test $ x > 1 $: Choose $ x = 2 $ (which is greater than 1) Substitute $ x = 2 $ into $ f''(x) $: $ f''(2) = 24(2) - 24 = 24 $ Since $ f''(2) > 0 $, the function is concave up on $ (1, \infty) $. Therefore, the function $ f(x) = 4x^3 - 12x^2 $ is concave down for $ x < 1 $ and concave up for $ x > 1 $.