Find the values of b for which y = 2x^4 - bx^2 has....
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skeeter
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for a polynomial function, inflection points occur where the second derivative equals zero and the sign of the second derivative changes.
y = 2x<sup>4</sup> - bx<sup>2</sup>
y' = 8x<sup>3</sup> - 2bx
y'' = 24x<sup>2</sup> - 2b = 2(12x<sup>2</sup> - b)
look at the equation 12x<sup>2</sup> - b = 0
if b < 0, then y'' cannot equal 0 ... no inflection points
if b = 0, then y'' = 0 at x = 0, but y'' > 0 for all nonzero values of x ... no inflection points
if b > 0, then y'' = 0 at x = +/- sqrt(b/12) and y" changes sign at each of those values ... two inflection points.
y = 2x<sup>4</sup> - bx<sup>2</sup>
y' = 8x<sup>3</sup> - 2bx
y'' = 24x<sup>2</sup> - 2b = 2(12x<sup>2</sup> - b)
look at the equation 12x<sup>2</sup> - b = 0
if b < 0, then y'' cannot equal 0 ... no inflection points
if b = 0, then y'' = 0 at x = 0, but y'' > 0 for all nonzero values of x ... no inflection points
if b > 0, then y'' = 0 at x = +/- sqrt(b/12) and y" changes sign at each of those values ... two inflection points.
The following is my solution in this question:-
d^2y/db^2 = 0 at point x = c
dy/db = 8x^3-2bx
d^2y/dx^2 = 24x^2 - 2b
2(12x^2 - b) = 0
12x^2 - b = 0
12x^2 = b
x^2 = b/12
Case 1: b>0
x = 0
Case 2: If b=0, the equation y=2x^4-bx^2 has 1 point of inflection.
Case 3: If b<0, the equation y=2x^4-bx^2 has 2 point of inflection.
Case 4: There is no answer.
Could you tell me the above solution correct or not?? Can you tell me the right answer if there is any problem?? Can you tell me how you type the mathematics symbols?? Do you want to install some of the software at your computer?? What about the name of it?
Cheers,
Tony
d^2y/db^2 = 0 at point x = c
dy/db = 8x^3-2bx
d^2y/dx^2 = 24x^2 - 2b
2(12x^2 - b) = 0
12x^2 - b = 0
12x^2 = b
x^2 = b/12
Case 1: b>0
x = 0
Case 2: If b=0, the equation y=2x^4-bx^2 has 1 point of inflection.
Case 3: If b<0, the equation y=2x^4-bx^2 has 2 point of inflection.
Case 4: There is no answer.
Could you tell me the above solution correct or not?? Can you tell me the right answer if there is any problem?? Can you tell me how you type the mathematics symbols?? Do you want to install some of the software at your computer?? What about the name of it?
Cheers,
Tony