Answer :

To find the inflection points, we first calculate the second derivative of the function and set it equal to zero. After finding the potential x-values, we verify the change in concavity around these points by checking the sign of the second derivative.

To find the inflection points of the function f(x) = 12x5 + 60x4 - 100x3 + 3, we need to find the second derivative of the function and set it equal to zero to solve for the x-values at which the concavity of the function changes.

Step 1: Find the first derivative f'(x)

f'(x) = 60x4 + 240x3 - 300x2

Step 2: Find the second derivative f''(x)

f''(x) = 240x3 + 720x2 - 600x

Step 3: Set the second derivative equal to zero

240x3 + 720x2 - 600x = 0

x(240x2 + 720x - 600) = 0

Let's solve for x:

x = 0

240x2 + 720x - 600 = 0 (Use the quadratic formula or factoring if possible)

Step 4: Verify the inflection points

We then need to confirm that the concavity actually changes at these x-values by analyzing the signs of the second derivative before and after these points.

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