# 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]}