differentiation

Question

1. Explain why the first order differential equates to zero for maximum and minimum value of a variable. With and illustration determine the maximum and minimum values of two related variables

Answer 1

Richard,

 

For any continuous function x over the real numbers, its first derivative or tangent function is also continuous. Roughly speaking, you don't need to lift your pencil off the paper to draw a continuous function since it has no instantaneous jumps in value that would cause its tangents from the right and left to be different at any point x.

 

As you approach from the left a local maximum of a function at x, its slope or first derivative approaching x is positive and then becomes negative to the right of x.  Being that the derivative is also continuous and goes from positive to negative, it had to pass through zero at point x.  Hence at the local extreme point of the function,its first derivative or slope is zero.

For a local minimum at x the argument is similar with the slope transitioning from negative to positive instead. Hence, also zero at the minimum point.

 

Example. Take y = sin x as the function where x is the angle in radians.  Being sinusoidal, the sin function has lots of local maximums and minimums.

Its first derivative dy/dx is cos x which is also sinusoidal but shifted in phase from the sin function.

 

From Geometry we know that the range of sin x is [-1, +1], and that:

sin x = +1 (local maximum) at x = Π/2 where cos x (its derivative) is zero (going from + to -).

sin x = -1 (local minimum) at x = 3Π/2 where cos x (its derivative) is zero (going from - to +).