Find the relative extrema for f(x, y) = 10^(xy).
So far, I have found the critical points by finding the partials with respect to x, and then with respect to y.
. . .Partial with respect to x: (ln10)(10^(xy))(y)
. . .Partial with respect to y: (ln10)(10^(xy))(x)
I then set the two equal to each other and tried to solve for x and y. Since 10^(xy) can never be zero, the only way for either expression to be zero is if x and y are themselves zero. Therefore, my critical point resulted in being (x, y) = (0, 0).
Then I had to classify the extrema using the 2nd partial test. The equation I used was:
. . .d = (fxx)(fyy)-[fxy]^2
I got everything (fxx, fxy, and fyy) to come out to be 0. When I plugged into d, I got the answer that the test was inconclusive. This doesn't seem right so I thought it wise to get a second opinion.
Thank you!
So far, I have found the critical points by finding the partials with respect to x, and then with respect to y.
. . .Partial with respect to x: (ln10)(10^(xy))(y)
. . .Partial with respect to y: (ln10)(10^(xy))(x)
I then set the two equal to each other and tried to solve for x and y. Since 10^(xy) can never be zero, the only way for either expression to be zero is if x and y are themselves zero. Therefore, my critical point resulted in being (x, y) = (0, 0).
Then I had to classify the extrema using the 2nd partial test. The equation I used was:
. . .d = (fxx)(fyy)-[fxy]^2
I got everything (fxx, fxy, and fyy) to come out to be 0. When I plugged into d, I got the answer that the test was inconclusive. This doesn't seem right so I thought it wise to get a second opinion.
Thank you!