Why the derivative is set equal to zero?

Answer 1

The first derivative is set to zero to find the critical points of the function. A critical point can be a minimum, maximum, or a saddle point. There's a reason for this. Suppose a differentiable function f:R->R has a maximum at x=a. Then the function goes down to the right of a, which means f'(a)<=0. Similarly, it also goes down to the left, so f'(a)>=0. Hence f'(a)=0. So the maxima and minima are all points where the derivative is 0. (But the converse isn't true; there could be points where the derivative is 0 which are not maxima or minima.) The derivative of a curve is its gradient. Setting the derivative to zero will find the points where the gradient is zero. If you look at the sketch of a curve you'll see that most interesting things happen where the gradient is zero. Around these points a curve can become a little 'snagged' or even change direction completely. Another point of view.. The derivative is set to zero to find the Boundary points. The Boundary could be the Highest Point/Maximum or the Lowest Point/minimum or both the highest and lowest point, a Saddle Point, highest in one direction and lowest in the other direction.