For what values of x is #f(x)=(x-1)(x-6)(x-2)# concave or convex?

Question

Emilia Aguilar · Accepted Answer

See below.

#f(x) = (x-1)(x6)(x-2) = x^3-9x^2+20x-12# #f'(x) = 3x^2-18x+20# #f''(x) = 6x-18# #f''(x)# is negative for #x < 3#, so #f# is concave (or concave down) on #(-oo,3)#. #f''(x)# is positive for #3 < x#, so #f# is convex (or concave up) on #(3,oo)#.

Aurora Alvarado · Answer

undefined To determine the concavity or convexity of the function $ f(x) = (x - 1)(x - 6)(x - 2) $, we need to examine the sign of its second derivative. First, compute the second derivative of $ f(x) $: $$ f''(x) = 2(x - 6) + 2(x - 1) + 2(x - 2) $$ $$ f''(x) = 2x - 12 + 2x - 2 + 2x - 4 $$ $$ f''(x) = 6x - 18 $$ For the function to be concave up (convex), $ f''(x) $ must be positive. For the function to be concave down (concave), $ f''(x) $ must be negative. Set $ f''(x) > 0 $ to find the intervals where the function is convex: $$ 6x - 18 > 0 $$ $$ x > 3 $$ So, the function is convex for $ x > 3 $. Set $ f''(x) < 0 $ to find the intervals where the function is concave: $$ 6x - 18 < 0 $$ $$ x < 3 $$ So, the function is concave for $ x < 3 $. Therefore, the function $ f(x) = (x - 1)(x - 6)(x - 2) $ is convex for $ x > 3 $ and concave for $ x < 3 $.