In my final exam review packet for my college calculus class, one of the problems is really giving me trouble.
The directions for the problem are: "Find all relative minimums and maximums for y= x^4 -2x^2+1 (use second derivative test)
The first thing I did was to find the first derivative, which was 4x(x²-1). Once you set that equal to zero you get critical points/numbers of 0, 1, -1.
From there I found the second derivative, 4(3x²-1).
I then took the critical points I found from the first derivative and put them on a number line to test on intervals with the second derivative. More specifically, I found test points between the critical numbers and tested those test points in the 2nd derivative function. Through that I found a relative Max at (-1,0) and a relative Min at (1,0).
I thought that was the answer, but the guide said different. It said the answer was: Rel. max (0,1) and rel. Min at (1,0) and (-1,0).
On a hunch, I tried another method and actually got the right answer. However, the method I used didn't include the second derivative at all.
What I did second was find the critical points just as I did before (0,1,-1), put them on a number line including test points just as before. But instead of testing using the second derivative, I used the first derivative function. Through that I got the same answer as in the answer guide.
That would be good, but the directions stated that I was to use the Second Derivative Test. What am I doing wrong?
The directions for the problem are: "Find all relative minimums and maximums for y= x^4 -2x^2+1 (use second derivative test)
The first thing I did was to find the first derivative, which was 4x(x²-1). Once you set that equal to zero you get critical points/numbers of 0, 1, -1.
From there I found the second derivative, 4(3x²-1).
I then took the critical points I found from the first derivative and put them on a number line to test on intervals with the second derivative. More specifically, I found test points between the critical numbers and tested those test points in the 2nd derivative function. Through that I found a relative Max at (-1,0) and a relative Min at (1,0).
I thought that was the answer, but the guide said different. It said the answer was: Rel. max (0,1) and rel. Min at (1,0) and (-1,0).
On a hunch, I tried another method and actually got the right answer. However, the method I used didn't include the second derivative at all.
What I did second was find the critical points just as I did before (0,1,-1), put them on a number line including test points just as before. But instead of testing using the second derivative, I used the first derivative function. Through that I got the same answer as in the answer guide.
That would be good, but the directions stated that I was to use the Second Derivative Test. What am I doing wrong?
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Your method eas perfect.
did you test and got answers as below ?
at x=0 y''=-4
at x=-1 y''=8
at x=1 y''=8
when value of y'' is -ve there is a maximum and when y'' is positive there is a minimum
hence at x=0 maximum and at x=-1 and x=1 minimum
did you test and got answers as below ?
at x=0 y''=-4
at x=-1 y''=8
at x=1 y''=8
when value of y'' is -ve there is a maximum and when y'' is positive there is a minimum
hence at x=0 maximum and at x=-1 and x=1 minimum
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Did you check at x = -1 ? The first derivative there is 0 and, as you know, and the second derivative is the same as x = 1. That seems like an ideal candidate.
Don't just test between the critical points, test on each side of the critical points.
See these page for more information::
http://www.wolframalpha.com/input/?i=+y%…
http://www.wolframalpha.com/input/?i=+pl…
http://www.wolframalpha.com/input/?i=+pl…
Did you check at x = -1 ? The first derivative there is 0 and, as you know, and the second derivative is the same as x = 1. That seems like an ideal candidate.
Don't just test between the critical points, test on each side of the critical points.
See these page for more information::
http://www.wolframalpha.com/input/?i=+y%…
http://www.wolframalpha.com/input/?i=+pl…
http://www.wolframalpha.com/input/?i=+pl…