Verified answer

To find the extrema, differentiate the function with respect to x. Set the derivative equal to zero. Solve for x.

f[x] = (x-1)^3 (x-3)

d(f[x])/dx = 3(x-1)^2 (x-3) + (x-1)^3

0 = 3(x-1)^2 (x-3) + (x-1)^3

0 = (x-1)^2 (3(x-3)+(x-1)) = (x-1)^2 (4x-10)

x = 1, 5/2

The derivative is equal to 0 at x=1 and x=5/2. This means that the slope of the original function is equal to 0 at x=1 and x=5/2. The slope can only be zero at local maxima, local minima, and inflection points.

Now take the derivative of the derivative (the second derivative of the original function with respect to x). Then plug in the values of x that we found above.

d(d(f[x])/d(dx) = 2(x-1)(4x-10)+4(x-1)^2

d(d(f[1])/d(dx) = 2(1-1)(4-10)+4(1-1)^2 = 0

At x=1, the slope of the second derivative is 0, which means x=1 is an inflection point (not an extrema).

d(d(f[5/2])/d(dx) = 2((5/2)-1)(4(5/2)-10)+4((5/2)-1)^2 = 9

At x=5/2, the slope of the second derivative is positive, which means x=5/2 is a local minima.