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Prove that (tan^(-1)1/e)^2+(2e)/((e^2+1)...

Prove that `(tan^(-1)1/e)^2+(2e)/((e^2+1)<(tan^(-1)e)^2+2/(sqrt(e^2+1))`

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ARIHANT MATHS-MONOTONICITY MAXIMA AND MINIMA-Exercise (Questions Asked In Previous 13 Years Exam)
  1. Prove that (tan^(-1)1/e)^2+(2e)/((e^2+1)<(tan^(-1)e)^2+2/(sqrt(e^2+1))

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  2. The least value of alpha in R for which 4alphax^(2)+(1)/(x)ge1, for al...

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  3. The number of points in (-oo,oo) for which x^(2)-xsinx-cosx=0, is

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  4. Let f : R ->(0,oo) and g : R -> R be twice differentiable functions...

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  5. Let f:(0,oo) in R be given f(x)=overset(x)underset(1//x)int e^-(t+(1...

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  6. The fuction f(x)=2|x|+|x+2|-||x+2|-2|x|} has a local minimum or a loca...

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  7. A rectagular sheet of fixed permeter with sides having thir lengths in...

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  8. A vertical line passing through the point (h, 0) intersects the ellips...

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  9. Let f,g and h be real-valued functions defined on the interval [0,1] b...

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  10. e total number of local maxima and local minima of the function f(x) =...

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  11. If the function g:(-oo,oo)->(-pi/2,pi/2) is given by g(u)=2tan^-1(e^u)...

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  12. The second degree polynomial f(x), satisfying f(0)=o, f(1)=1,f'(x)gt...

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  13. If f(x)=x^(3)+bx^(2)+cx+dand 0 ltb^(2)ltc, then in (-oo,oo)

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  14. If f(x)=x^(2)+2bx+2c^(2)and g(x)=-x^(2)-2cx+b^(2), such that minf(x)gt...

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  15. The length of a longest interval in which the function 3sinx-4sin^(3...

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  16. If f(x)=e^(1-x) then f(x) is

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  17. The maximum value of (cosalpha(1))-(cos alpha(2))...(cosalpha(n)), u...

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  18. If f(x) = {{:(e ^(x),,"," 0 le x lt 1 ,, ""), (2- e^(x - 1),,"," 1 lt ...

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  19. If f(x) is a cubic polynomil which as local maximum at x=-1 . If f(2)=...

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  20. Consider the function f:(-oo, oo) -> (-oo ,oo) defined by f(x) =(x^2...

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  21. Consider the function f:(-oo, oo) -> (-oo ,oo) defined by f(x) =(x^2...

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