Question

Find the largest open intervals on which the function is concave upward or concave downward, and find the location of any points of inflection. f(x)=2x³+6x²−8x+2

Answer 1

To find the intervals on which the function is concave upward or concave downward, we need to examine the second derivative of the function.

Step-by-step explanation:

To find the intervals on which the function is concave upward or concave downward, we need to examine the second derivative of the function.

To do this, we first find the first derivative of the function:
f'(x) = 6x² + 12x - 8

Next, we find the second derivative:
f''(x) = 12x + 12

Setting the second derivative equal to zero and solving for x, we get:
12x + 12 = 0
x = -1

So, the function changes concavity at x = -1.

Now we need to test the intervals between the critical points and the endpoints of the domain.

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