Question

determine where the graph is concave up or down y=-x^3+3x^2-2

Answer 1

well lots of ways to look at this

easiest if to find the derivative... are you able to do that..?

Answer 2

derivative is -3x^2+6x

Answer 3

setting thhat equal to 0 I get x=0 and x=2

Answer 4

ok so they are the 2 stationary points...

find the 2nd derivative and test the stationary points

\[\frac{d^2y}{dx^2} > 0 ....minimum..... \frac{d^2y}{dx^2} < 0 .... maximum \]

Answer 5

\[\frac{d^2y}{dx^2} = -6x + 6\]

so a point of inflection occurs when the 2nd derivative = 0

so a point of inflexion is x = 1

Answer 6

so what did you find...?

by substituting x = 0 and x = 2 into the 2nd derivative...

Answer 7

so the curve has 2 parts... one is concave up... the other is concave down... than the change in concavity occurs at x = 1

Answer 8

plugging in 0 I got 6, plugging in 2 I got -6

Answer 9

ok... so the curve is concave up at x = 0, concave down at x = 2



the change in concavity occurs at x = 1

hope this makes sense.