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.
