If roots of the equation $x^4-8x^3+bx^2+cx+16=0$ are positive, then
- $b=c=8$
- $b=-24,c=-32$
- $b=24,c=-32$
- $b=24,c=32$
For a quartic equation with positive roots, by Vieta's formulas, the product of roots is 16 and the sum is 8. If all roots are equal (x=2), the equation is (x-2)^4 = x^4 - 8x^3 + 24x^2 - 32x + 16 = 0. Comparing coefficients, b=24 and c=-32.
By Vieta's formulas for the equation x^4 - 8x^3 + bx^2 + cx + 16 = 0, the sum of the four positive roots is 8 and their product is 16. By the AM-GM inequality, the sum of the roots divided by 4 is greater than or equal to the fourth root of their product, giving 8/4 >= (16)^(1/4), which simplifies to 2 >= 2. This equality means all four positive roots must be exactly 2. Therefore, the sum of products of roots taken two at a time is (4 choose 2)*2*2 = 24, so b = 24, and the sum of products of roots taken three at a time is 4*2*2*2 = 32, but applied to the equation gives -c = 32, meaning c = -32. The result is b = 24, c = -32.