How do you determine the intervals on which function is concave up/down & find points of inflection for #y=4x^5-5x^4#?

1 Answer
Oct 23, 2015

Answer:

Investigate the sign of the second derivative.

Explanation:

#y=4x^5-5x^4#

#y'=20x^4-20x^3 = 20(x^4-x^3)#

#y'' =20(4x^3-3x^2) = 20x^2(4x-3)#

#y''# is never undefined and #y''=0# at #x=0# and at #x=3/4#.

Because #20x^2# is always positive, the sign of #y''# is the same as the sign of #4x-3# (or build a sign table of sign diagram or whatever you have learned to call it, for #y''#).

#y''# is negative (so the graph of the function is concave down, for #x<3/4# and

#y''# is posttive (so the graph of the function is concave up, for #x > 3/4#

The curve is concave down on the interval #(-oo,3/4)# and
concave up on #(3/4,oo)#.

The int #(3/4, -81/128)# is the only inflection point.