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Calculus Level 5

Let ff be a double differentiable function and satisfy the condition f(0)=0,f(1)=0f(0)=0, f(1)=0 and d2dx2(exf(x)x2)>0  x(0,1)\dfrac{d^2}{dx^2}\left (e^{-x}f(x)-x^2\right )>0\ \forall\ x\in (0,1)

Then the sum of values of x(0,1)x\in (0,1) such that f(x)3=(x2x)exf(x)-3=(x^2-x)e^x is ϕ\phi

The number of ordered pair(s) (x,y)(x,y) of real numbers satisfying equation 1+x4+2x2sin(cos1y)=01+x^4+2x^2\sin(\cos^{-1}y)=0 is ζ\zeta.

Calculate ζ+ϕ\zeta+\phi.


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