Question

Define each of the following for a function of two variables. (a) Relative minimum (b) Relative maximum (c) Saddle point (d) Critical point

Short answer

Relative minimum: point where function's value is less or equal to all nearby points. Relative maximum: point where function's value is greater or equal to all nearby points. Saddle point: critical point that is neither a relative minimum nor maximum. Critical point: point where derivative is zero or undefined.

Step-by-step solution

Define Relative Minimum

A relative minimum of a function of two variables occurs at a point if the value of the function at that point is less than or equal to the values of the function at all nearby points. In mathematical terms, for a function \(f(x, y)\), point \((a, b)\) is a relative minimum if \(f(a, b) \leq f(x, y)\) for all points \((x, y)\) within some region around \((a, b)\).

Define Relative Maximum

A relative maximum of a function of two variables occurs at a point if the value of the function at that point is greater than or equal to the values of the function at all nearby points. In mathematical terms, for a function \(f(x, y)\), point \((a, b)\) is a relative maximum if \(f(a, b) \geq f(x, y)\) for all points \((x, y)\) within some region around \((a, b)\).

Define Saddle Point

A saddle point of a function of two variables is a critical point that is neither a relative minimum nor a relative maximum. In other words, it is a point where the function is not locally convex or concave; it bones up in some directions and down in others. This kind of point is called a saddle point because in three dimensions it looks like a saddle.

Define Critical Point

A critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0 or undefined. For a function of two variables \(f(x, y)\), a critical point is any point \((a, b)\) in the domain where the gradient \(\nabla f\) is either zero or undefined.