Exercise · ZW-CRITICAL-POINT-MULTIVARIABLE-010 · ★★★ Advanced
Let \(f(x,y)=x^2-y^2\). Which statement is true?
Choose one option.
Show answer
- C. \((0,0)\) is a critical point but not a local maximum or minimum
Explanation
\(\nabla f=\langle2x,-2y\rangle\) vanishes at the origin, but \(f(x,0)=x^2>0\) and \(f(0,y)=-y^2<0\): it is not an extremum. Being critical is necessary, not sufficient.
Review the concept: Critical point of a multivariable function
A critical point of a function \(f\) of two variables is a point \((a,b)\) of its domain where the gradient \(\nabla f\) is the zero vector or does not exist.
\[\nabla f(a,b)=\mathbf 0\quad\text{or}\quad\nabla f(a,b)\text{ does not exist}\]
More practice problems: critical point of a multivariable function
- ZW-CRITICAL-POINT-MULTIVARIABLE-001 Find the critical point of f(x,y)=x^2+y^2-4x+6y.
- ZW-CRITICAL-POINT-MULTIVARIABLE-002 Which of these points is critical for f(x,y)=x^3-3x+y^2?
- ZW-CRITICAL-POINT-MULTIVARIABLE-003 Find the critical point of f(x,y)=x^2+xy+y^2-3x.
- ZW-CRITICAL-POINT-MULTIVARIABLE-004 Find the critical point of f(x,y)=xy-x-2y.
- ZW-CRITICAL-POINT-MULTIVARIABLE-005 How many critical points does f(x,y)=x^3-3x+y^3-3y have?
- ZW-CRITICAL-POINT-MULTIVARIABLE-006 How many critical points does f(x,y)=x^2+y^4 have?
- ZW-CRITICAL-POINT-MULTIVARIABLE-007 Let (a,b) be the critical point of f(x,y)=x^2+y^2-2x-4y+7. Compute f(a,b).
- ZW-CRITICAL-POINT-MULTIVARIABLE-008 For which value of k is (1,0) critical for f(x,y)=x^2+y^2+kx?