Find all the values of k such that f(x) = x^3 - 3x^2 + k has three different x-intercepts.?

Plot the curve f(x) = x^3 -3x^2. Red line below.

Notice that there are 2 x intercepts. If you raise the curve upward by a value less than the distance from the x axis to f(2), you will have three intercepts.

So make your value of k...

0 < k < 4

The blue line represents k = 2

graphically method --> 0<k<4

This will be where the x axis is always between local max and local min of the function.

This is between the two values of x where the derivative equals zero.

dy/dx = 3x^2 - 6x = 0

3x^2 = 6x

x^2 = 2x

x = 2 or x = 0

f(0) = 0+k

f(2) = 8-12+k = -4+k

So 0 < k < 4