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Given y = sin²(x) + cos(x), and d²x/dy² evaluated at x = pi/4 equals a + b*sqrt(2) where a, b are integers, find (a + b + 2)/13.

1. 0
2. 1
3. 2
4. 3

Correct answer: 2

Solution

y = sin² x + cos x. dy/dx = 2 sin x cos x - sin x = sin x(2cos x - 1). At x=pi/4: dy/dx = (sqrt(2)/2)(sqrt(2)-1) = 1 - 1/sqrt(2) = (sqrt(2)-1)/sqrt(2). d²y/dx² = 2cos(2x) - cos x. At x=pi/4: d²y/dx² = 2*0 - 1/sqrt(2) = -1/sqrt(2). d²x/dy² = -(d²y/dx²)/(dy/dx)³ = (1/sqrt(2)) / ((sqrt(2)-1)³ / (2sqrt(2))^(3/2))... Simplify (dy/dx)³ = ((sqrt(2)-1)/sqrt(2))³ = (sqrt(2)-1)³ / (2sqrt(2)). (sqrt(2)-1)³ = 5sqrt(2)-7. So (dy/dx)³ = (5sqrt(2)-7)/(2sqrt(2)). d²x/dy² = (1/sqrt(2)) / ((5sqrt(2)-7)/(2sqrt(2))) = (1/sqrt(2))*(2sqrt(2)/(5sqrt(2)-7)) = 2/(5sqrt(2)-7). Rationalising: 2(5sqrt(2)+7)/((50-49)) = 2(5sqrt(2)+7) = 10sqrt(2)+14. So a=14, b=10. (a+b+2)/13 = 26/13 = 2.

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