In one line
A function \(f(x,y)\) has a maximum (minimum) at a point if its value there is larger (smaller) than at all nearby points; such points are found among the stationary points where \(f_x=0\) and \(f_y=0\).
Explained simply
Picture a landscape of hills and valleys, where the height above sea level is \(f(x,y)\). The top of a hill is a maximum, the bottom of a valley is a minimum. At both places the ground is flat: walking a tiny step east or north does not change your height at first. That flatness is the condition \(f_x=0\) and \(f_y=0\).
Technically, \(f(a,b)\) is a (local) maximum if \(f(a+h,b+k)<f(a,b)\) for all small \(h,k\) not both zero, and a (local) minimum if \(f(a+h,b+k)>f(a,b)\). The necessary condition for either is \(f_x(a,b)=0\) and \(f_y(a,b)=0\). Points satisfying this are called stationary (or critical) points. A stationary point is not automatically a maximum or minimum; it may be a saddle point, so a second test is needed.
How it works
- Find \(f_x\) and \(f_y\).
- Solve the simultaneous equations \(f_x=0\), \(f_y=0\). Factorise where possible and list every solution pair \((x,y)\).
- These are the stationary points. Classify each one using the second derivative test (\(rt-s^2\)).
- Substitute the point into \(f\) to get the maximum or minimum value.
Key formula
Stationary point conditions: \(\dfrac{\partial f}{\partial x}=0,\ \dfrac{\partial f}{\partial y}=0\). The extreme value is \(f(a,b)\), in the same units as \(f\).
Quick example
\(f(x,y)=x^2+y^2-4x-6y+20\).
\(f_x=2x-4=0\Rightarrow x=2\); \(f_y=2y-6=0\Rightarrow y=3\). Only stationary point: \((2,3)\).
Value: \(f(2,3)=4+9-8-18+20=7\). Completing the square, \(f=(x-2)^2+(y-3)^2+7\ge7\), so 7 is the minimum. The surface is a bowl with its lowest point at \((2,3)\).
Common mistakes
- Missing stationary points when solving, e.g. dividing by \(x\) and losing \(x=0\).
- Declaring every stationary point a maximum or minimum without testing.
- Giving only the point and not the maximum or minimum value of \(f\).
How MU asks it
The syllabus gives this 1 hour, so expect one 5/6-mark question: "Find the stationary values of \(x^3+y^3-3axy\), \(a>0\)" or "Examine \(f(x,y)=x^3+3xy^2-15x^2-15y^2+72x\) for extreme values". For full marks: show \(f_x,f_y\), solve for all stationary points, compute \(r,s,t\) at each, classify, and state the extreme values.