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
Jim H
Oct 23, 2015

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.

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