The critical point can be a maximum, minimum or neither

• slope changes from increasing to decreasing, point is a local maximum
• slope changes from decreasing to increasing, point is a local minimum

If $f$ is differentiable at $x=a$ that attains a maximum or minimum value at $x=a$, then $f^{\prime }(a)=0$

If $f$ is twice differentiable at a critical point $(a,f(a))$ then:

• if $f^{″}(a)>0\to (a,f(a))$ is a local minimum
• if $f^{″}(a)<0\to (a,f(a))$ is a local maximum

Multi-Variable

If $f(x,y)$ has continuous first partial derivatives and $(a,b)$ is a local maximum or minimum of $f(x,y)$ then $f_x(a,b)=0$ and $f_y(a,b)=0$

Hessian matrix

Second derivative test

Suppose that $f(x,y)$ has continuous second partial derivatives near$(a,b)$ and $f_x(a,b)$ $=f_y(a,b)=0$ (critical point)

If $Hf(a,b)$ is

1. positive definite then $(a,b)$ is a local minimum of $f$

• det$(Hf(a,b))>0,f_{xx}(a,b)>0$

2. negative definite then $(a,b)$ is a local maximum of $f$

• det($Hf(a,b))>0,f_{xx}(a,b)<0$

3. indefinite then $(a,b)$ is a saddle point of $f$

• $det(Hf(a,b))<0$