Differentiation - Maximum and Minimum

The gradient at the maximum and minimum point is 0. Thus, to find maximum and minimum, these are the steps required.

Step 1: If you need to find maximum or minimum value of V, you first need an equation involving V.

Step 2: Differentiate the equation, to find e.g. dV/dx.

Step 3: Equate dV/dx to 0, and find x.

Step 4: Substitute x into equation of V (from step 1) to find the stationary value of V.

Step 5: To determine maximum or minimum, differentiate dV/dx to find d²V/dx². Substitute in x from step 4.
If d²V/dx² > 0, V is minimum.

If d²V/dx² < 0, V is maximum.

Example question:
Find the maximum volume of the tank, if its volume, V, is given by the following equation:

Answers: