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
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.
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)
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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