To find minimum or maximum force for friction problems in physics, we will need to use the knowledge in calculus. No matter the force we are looking for is minimum or maximum, we can just take the first derivative of the force and set its value to zero. Why? That's because at the minimum or maximum value of the force, the slope of the tangent line at that point is ALWAYS zero. (You may remember in caculus the slope of the tangent line is equal to the first derivative of the function, and that's why we set it to zero, so slope is zero.)

But if the question asks you whether the force is at its maximum or minimum, first derivative is not enough to help. You only find the min/max value using first derivative, but you don't know the value you find is minimum or maximum. In approach problems like that, you need to find second derivative: If the second derivative is POSITIVE, then the force is at its MINIMUM; If the second derivative is NEGATIVE, then the force is at its MAXIMUM.

Still have no idea what I am talking about because you forget the Caclulus you learnt? Then please read my other article (link provided below) discussing the properties of first derivative and second derivative:

Relationship between f, f' and f''