For what values of x is #f(x)= -2x^3 + 4 x^2 + 5x -5 # concave or convex?

1 Answer
Dec 10, 2017

Concave: #{x in RR: 2/3< x < oo}#

Convex: #{x in RR: -oo < x < 2/3 }#

Explanation:

A function is convex if the second derivative is positive and concave where its second derivative is negative. When the second derivative is #0# this could mean the function is concave , convex, or it could be a point of inflexion. This would have to be tested using the first derivative.

#f(x)=-2x^3+4x^2+5x-5#

#dy/dx(-2x^3+4x^2+5x-5)=-6x^2+8x+5#

#(d^2y)/(dx^2)(-6x^2+8x+5)=-12x+8#

So:

#-12x+8<0#

#x>8/12#

#x >2/3#

Concave: #{x in RR: 2/3< x < oo}#

#-12x+8>0#

#x<8/12#

#x<2/3#

Convex: #{x in RR: -oo < x < 2/3 }#

Graph:

graph{y=-2x^3+4x^2+5x-5 [-4, 4, -7.12, 7.12]}