# How do I use the second derivitive test to find the points of inflection and discuss the concavity of this funciton: y= 1/2x^4 + 2x^3

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How do I use the second derivitive test to find the points of inflection and discuss the concavity of this funciton:

y= 1/2x^4 + 2x^3

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Step 1

Let's proceed to find f'(x) and f''(x) and then find the point of inflection by equating f''(x) to 0. Please see the white  board.

Step 2

Let's set f''(0) to get the point of inflection. Please see the white board.

f(0) = 0; f(-2) = -8

Hence, the points of inflection are (-2, -8) and (0, 0)

Step 3

f'(x) = 6x2  + 12x = 6x(x + 2)

If f’’(x) > 0;  x(x+2) > 0 Hence:

Case 1: x > 0 and x + 2 > 0 i.e x > 0

Case 2: x < 0 and x + 2 < 0 i.e. x < -2

The function is concave upward...

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