How do I find the maximum/minimum of a curve?

To find the maximum/minimum of a curve you must first differentiate the function and then equate it to zero. This gives you one coordinate. To find the other you must resubstitute the one already found into the original function. To determine whether the point on the curve is a maximum or minimum differentiate to the second order and substitute a coordinate in. If the value is positive it is a minimum point & vice versa.

Example: Find the coordinates of the maximum of the curve y=6x^1/2-x-3 

y=6x^1/2​-x-3 

dy/dx=3x^-1/2 -1  d²y/dx²=-3/2x^-3/2

3x^1/2 ​-1=0 

x=9 therefore y=6

Sub x=6 into  d²y/dx² to give -1/18 so its a maximum point with coordinates (9,6)