Options are 1) Local maximum 2) Local minimum 3) Saddle 4) Cannot be classified

Question

Options are

1) Local maximum

2) Local minimum

3) Saddle

4) Cannot be classified

Consider the following function: This function has one critical point at p=(0,0). If possible,classify the critical point using the second-derivative test in each of the following . If both a and c are positive, then at the point p, f has: If a is positive but c is negative, then at the point p, f has: If a is negative but c is positive, then at the point p, f has: If both a and c are negative, then at the point p, f has:

Explanation / Answer

f(x,y)=(1/2)ax2+(1/2)cy2

Then,

fx(x,y)=ax, and so

fxx(x,y)=a

and

fxy(x,y)=0

similarly

fyy(x,y)=c

Now we need to look at:

D(0,0)=fxx(0,0)fyy(0,0)-(fxy(0,0))^2=ac-0=ac

Thus,

if a and c are positive then so is D(0,0) as well as fxx(0,0). Hence there is a:

local minimum

If a is positive but c is negative, then D(0,0) is negative. Hence there is a:

saddle point

If a is negative but c is positive, then D(0,0) is negative. Hence there is a:

saddle point

If both a and c are negative, then D(0,0) is positive. Also, fxx(0,0) is negative, hence there is a:

local maximum

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