From 1-D to 2-D

Calculus Level 4

The extrema test states that for a function f : R → R f: \mathbb{R} \rightarrow \mathbb{R} , if a point x ∗ x^* satisfies
∂ f ∂ x = 0 , ∂ 2 f ∂ x 2 > 0 , \frac{ \partial f } { \partial x } = 0, \quad \frac{ \partial^2 f } { \partial x^2 } > 0,
then we have a local maximum.

True or False?

For a function f : R × R → R f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} , if a point ( x ∗ , y ∗ ) (x^*, y^*) satisfies
∂ f ∂ x = 0 , ∂ f ∂ y = 0 , ∂ 2 f ∂ x 2 > 0 , ∂ 2 f ∂ y 2 > 0 , \frac{ \partial f } { \partial x } = 0, \frac{ \partial f } { \partial y } = 0, \quad \frac{ \partial^2 f } { \partial x^2 } > 0, \frac{ \partial^2 f } { \partial y^2 } > 0,
then we have a local maximum.

True, it must be a local maximum False, it could be a saddle point False, it could be an inflection point False, it could be a local minimum

Calvin Lin by Calvin Lin

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1 solution

梦 叶
Dec 7, 2016

the first derivatives of x and y are 0, point (x ,y ) can be minimum, maximum or saddle point. We need to calculate:

N1 = f xx and N2 = f xxf yy-f xy.

If N1 and N2 both are greater then 0, we have a minimum.

If N1<0 and N2 >0, we have a maximun.

If N1 and N2 both are less than 0, we have a saddle point.

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