For what values of x is #f(x)= x^3 - 4x^2 + 5x # concave or convex?
1 Answer
May 7, 2016
concave
convex
Explanation:
To determine where f(x) is concave/convex we require to find f''(x)
f(x)
#=x^3-4x^2+5x# f'(x)
#=3x^2-8x+5# and f''(x)
#=6x-8# We now equate f''(x) to zero to find values of x where any change from concave/convex or convex/concave may occur.
solve : 6x - 8 = 0
#rArr x=4/3# We now have to check the value of f''(x) to the left and right of
#x=4/3 , "say " x=a# • If f''(a) > 0 , then f(x) is convex
• If f''(a) < 0 , then f(x) is concave
x = 0 is to the left and f''(0) = - 8 → concave
x = 2 is to the right and f''(2) = 4 → convex
#"hence" f(x)" is concave " (-oo,4/3)# and f(x)
#" is convex " (4/3,+oo)#
graph{x^3-4x^2+5x [-10, 10, -5, 5]}
