Answer to Question #97141 in Calculus for Anand

Question

determine the points of inflection of the curve, y= x⁴–4x³–18x²+1, if any.

Answer 1

Determine the points of inflection of the curve, y= x⁴–4x³–18x²+1, if any.

Solution:

Take the first derivative: y' = 4x³- 12x²-36x.

Take the second derivative: y'' = 12x²- 24x-36;

y'' = 12x²- 24x-36=0.

Let`s solve the last equation x₁=3 and x₂= -1.

The expression 12x²- 24x-36 is negative for -1<x<3 , positive for x<-1 or x>3. So:

y is concave downward if −1<x<3;

y is concave upward if x < −1 or x > 3.

Thus, the inflection points are at x = −1 and x = 3.

So we have two points of inflection, namely, (3,-188) and (-1,-12).