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Question Number 2548 by Filup last updated on 22/Nov/15
For a function y=f(x), inflection points/stationary points are when (df/dx)=0. For a function z=f(x, y), can you find these points through a similar method? Is it something like (∂f/∂x)=0 and (∂f/∂y)=0?
$$\mathrm{For}\:\mathrm{a}\:\mathrm{function}\:{y}={f}\left({x}\right), \\ $$$$\mathrm{inflection}\:\mathrm{points}/\mathrm{stationary}\:\mathrm{points}\:\mathrm{are} \\ $$$$\mathrm{when}\:\:\frac{{df}}{{dx}}=\mathrm{0}. \\ $$$$ \\ $$$$\mathrm{For}\:\mathrm{a}\:\mathrm{function}\:{z}={f}\left({x},\:{y}\right),\:\mathrm{can}\:\mathrm{you}\:\mathrm{find} \\ $$$$\mathrm{these}\:\mathrm{points}\:\mathrm{through}\:\mathrm{a}\:\mathrm{similar}\:\mathrm{method}? \\ $$$$ \\ $$$$\mathrm{Is}\:\mathrm{it}\:\mathrm{something}\:\mathrm{like}\:\frac{\partial{f}}{\partial{x}}=\mathrm{0}\:\mathrm{and}\:\frac{\partial{f}}{\partial{y}}=\mathrm{0}? \\ $$
Answered by Yozzi last updated on 22/Nov/15
Let f be a function of two variables whose first and second partial derivatives are continous on some open disc B. Suppose further that at the point (a,b) f_x =(∂f/∂x)=0 and f_y =(∂f/∂y)=0. Let p=(∂^2 f/∂x^2 )=f_(xx) , q=(∂^2 f/∂y^2 )=f_(yy) and r=(∂^2 f/(∂y∂x))=f_(xy) (a,b) is a local minimum if pq−r^2 >0 and p>0 (or q>0), (a,b) is a local maximum if pq−r^2 >0 and p<0 (or q<0) (a,b) is a saddle point if pq−r^2 <0 pq−r^2 =0 is inconclusive.
$${Let}\:{f}\:{be}\:{a}\:{function}\:{of}\:{two}\:{variables} \\ $$$${whose}\:{first}\:{and}\:{second}\:{partial}\: \\ $$$${derivatives}\:{are}\:{continous}\:{on}\:{some}\:{open} \\ $$$${disc}\:{B}.\:{Suppose}\:{further}\:{that}\:{at}\:{the}\:{point} \\ $$$$\left({a},{b}\right) \\ $$$${f}_{{x}} =\frac{\partial{f}}{\partial{x}}=\mathrm{0}\:\:\:{and}\:\:\:{f}_{{y}} =\frac{\partial{f}}{\partial{y}}=\mathrm{0}.\: \\ $$$${Let}\:{p}=\frac{\partial^{\mathrm{2}} {f}}{\partial{x}^{\mathrm{2}} }={f}_{{xx}} \:,\:{q}=\frac{\partial^{\mathrm{2}} {f}}{\partial{y}^{\mathrm{2}} }={f}_{{yy}} \:{and}\:{r}=\frac{\partial^{\mathrm{2}} {f}}{\partial{y}\partial{x}}={f}_{{xy}} \\ $$$$\left({a},{b}\right)\:{is}\:{a}\:{local}\:{minimum}\:{if}\:{pq}−{r}^{\mathrm{2}} >\mathrm{0}\:\:{and}\:{p}>\mathrm{0}\:\left({or}\:{q}>\mathrm{0}\right), \\ $$$$\left({a},{b}\right)\:{is}\:{a}\:{local}\:{maximum}\:{if}\:{pq}−{r}^{\mathrm{2}} >\mathrm{0}\:\:\:{and}\:{p}<\mathrm{0}\:\:\left({or}\:\:{q}<\mathrm{0}\right) \\ $$$$\left({a},{b}\right)\:{is}\:{a}\:{saddle}\:{point}\:{if}\:{pq}−{r}^{\mathrm{2}} <\mathrm{0} \\ $$$${pq}−{r}^{\mathrm{2}} =\mathrm{0}\:{is}\:{inconclusive}. \\ $$
Commented by Filup last updated on 22/Nov/15
My question is what is an open disc and what is f_(xy) ? is (∂^2 f/(∂x∂y)) differentiation with two variables?
$$\mathrm{My}\:\mathrm{question}\:\mathrm{is} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{an}\:\mathrm{open}\:\mathrm{disc}\:\mathrm{and}\:\mathrm{what}\:\mathrm{is}\:{f}_{{xy}} ? \\ $$$$\mathrm{is}\:\:\frac{\partial^{\mathrm{2}} {f}}{\partial{x}\partial{y}}\:\:\mathrm{differentiation}\:\mathrm{with}\:\mathrm{two}\:\mathrm{variables}? \\ $$
Commented by Yozzi last updated on 22/Nov/15
f_x is partial differentiation of f w.r.t x f_(xy) =(∂/∂y)(f_x )=(∂/∂y)((∂f/∂x))=(∂^2 f/(∂y∂x)) (mixed 2nd partial derivative) An n−dimensional open disc of radius r is the collection of points less than a distance of r away from a fixed point in Euclidean n− space. (Wolfram Alpha) If n=1 for example, this is an open interval (s,t). So, for n=2 this defines a collection of points in a circle containing the point (a,b) where both f_x and f_(y ) are zero. In this case, once the 1st and 2nd partial derivatives of f exist and the both f_(xy) and f_(yx ) are continuous for the set of points (x,y) in the disc, these two mixed partial 2nd derivatives are equal, i.e f_(xy) =f_(yx) .
$${f}_{{x}} \:{is}\:{partial}\:{differentiation}\:{of}\:{f}\:{w}.{r}.{t}\:\:{x} \\ $$$${f}_{{xy}} =\frac{\partial}{\partial{y}}\left({f}_{{x}} \right)=\frac{\partial}{\partial{y}}\left(\frac{\partial{f}}{\partial{x}}\right)=\frac{\partial^{\mathrm{2}} {f}}{\partial{y}\partial{x}}\:\left({mixed}\:\mathrm{2}{nd}\:{partial}\:{derivative}\right) \\ $$$${An}\:{n}−{dimensional}\:{open}\:{disc}\:{of} \\ $$$${radius}\:{r}\:{is}\:{the}\:{collection}\:{of}\:{points}\:{less} \\ $$$${than}\:{a}\:{distance}\:{of}\:{r}\:{away}\:{from}\:{a}\:{fixed} \\ $$$${point}\:{in}\:{Euclidean}\:{n}−\:{space}.\:\left({Wolfram}\:{Alpha}\right) \\ $$$${If}\:{n}=\mathrm{1}\:{for}\:{example},\:{this}\:{is}\:{an}\:{open}\: \\ $$$${interval}\:\left({s},{t}\right).\:{So},\:{for}\:{n}=\mathrm{2}\:{this}\: \\ $$$${defines}\:{a}\:{collection}\:{of}\:{points}\:{in}\:{a}\:{circle}\:{containing} \\ $$$${the}\:{point}\:\left({a},{b}\right)\:{where}\:{both}\:{f}_{{x}} \:{and}\:{f}_{{y}\:} \\ $$$${are}\:{zero}.\:{In}\:{this}\:{case},\:{once}\:{the}\:\mathrm{1}{st} \\ $$$${and}\:\mathrm{2}{nd}\:{partial}\:{derivatives}\:{of}\:{f}\:{exist} \\ $$$${and}\:{the}\:{both}\:{f}_{{xy}} \:{and}\:{f}_{{yx}\:\:} \:{are} \\ $$$${continuous}\:{for}\:{the}\:{set}\:{of}\:{points}\:\left({x},{y}\right) \\ $$$${in}\:{the}\:{disc},\:{these}\:{two}\:{mixed}\:{partial} \\ $$$$\mathrm{2}{nd}\:{derivatives}\:{are}\:{equal},\:{i}.{e}\:{f}_{{xy}} ={f}_{{yx}} . \\ $$
Commented by Filup last updated on 22/Nov/15
I see. Very interesting. I must go learn multi−variable calculus!
$$\mathrm{I}\:\mathrm{see}.\:\mathrm{Very}\:\mathrm{interesting}.\:\mathrm{I}\:\mathrm{must}\:\mathrm{go}\:\mathrm{learn} \\ $$$$\mathrm{multi}−\mathrm{variable}\:\mathrm{calculus}! \\ $$

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