For what values of x is #f(x)=(x+2)(x-3)(x+1)# concave or convex?

Question

Scarlett Allison · Accepted Answer

The function is concave for #x in (-oo,0)#
The function is convex for #x in (0,+oo)# We need #(uvw)'=u'vw+uv'w+uvw'# We must calculate the second derivative and determine the sign. #f(x)=(x+2)(x-3)(x+1)# #f'(x)=(x-3)(x+1)+(x+2)(x+1)+(x+2)(x-3)# #=x^2-2x-3+x^2+3x+2+x^2-x-6# #=3x^2-7# #f''(x)=6x# Therefore, #f''(x)=0# when #x=0# This is the point of inflexion. We can build a chart #color(white)(aaaa)##Interval##color(white)(aaaaaa)##(-oo,0)##color(white)(aaaaaa)##(0,+oo)# #color(white)(aaaa)##sign f''(x)##color(white)(aaaaaaa)##-##color(white)(aaaaaaaaaaa)##+# #color(white)(aaaa)## f(x)##color(white)(aaaaaaaaaaaaa)##nn##color(white)(aaaaaaaaaaa)##uu#

Christopher Allers · Answer

undefined To determine the concavity of $ f(x) = (x + 2)(x - 3)(x + 1) $, we need to examine its second derivative. If the second derivative is positive, the function is concave upward (convex), and if it's negative, the function is concave downward.

First, find the first derivative:

$$ f'(x) = (x - 3)(x + 1) + (x + 2)(x + 1) + (x + 2)(x - 3) $$

Then, find the second derivative:

$$ f''(x) = 2(x - 3) + 2(x + 1) + 2(x + 2) $$

$$ f''(x) = 2x - 6 + 2x + 2 + 2x + 4 $$

$$ f''(x) = 6x $$

Since the coefficient of $ x $ in the second derivative $ 6x $ is positive for all real values of $ x $, the function $ f(x) $ is concave upward (convex) for all $ x $.