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

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

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To prove that \[ \left(\tan^{-1}\left(\frac{1}{e}\right)\right)^{2} + \frac{2e}{\sqrt{e^{2}+1}} < \left(\tan^{-1}(e)\right)^{2} + \frac{2}{\sqrt{e^{2}+1}}, \] we will define a function and analyze its monotonicity. ...
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CENGAGE ENGLISH-MONOTONICITY AND MAXIMA MINIMA OF FUNCTIONS-Solved Examples
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  2. Of f(x)=2/(sqrt(3))tan^(-1)((2x+1)/(sqrt(3)))-log(x^2+x+1)(lambda^2-5l...

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  5. Using the relation 2(1−cosx)<x^2 ,x=0 or prove that sin(tanx)≥x,∀ϵ[0,...

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  6. lf f'(x) > 0,f"(x)>0AA x in (0,1) and f(0)=0,f(1)=1,then prove that...

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

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  9. Prove that sin^2thetathetasin(sintheta)for0<thetatheta<pi

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  10. Let f(x) = 1 + 4x - x^(2), AA x in R g(x) = max {f(t), x le t le (x ...

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  11. Show that 5x le 8 cos x-2cos 2x le 6x for x le x le (pi)/(3)

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  12. Let f(x),xgeq0, be a non-negative continuous function. If f^(prime)(x)...

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  13. If a x^2+b/xgeqc for all positive x where a >0 and b >0, show that 27 ...

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  14. Prove that for x in [0, (pi)/(2)], sin x + 2x ge (3x(x + 1))/(pi).

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  15. Consider a curve C : y=cos^(-1)(2x-1) and a straight line L :2p x-4y+2...

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  16. From a fixed point A on the circumference of a circle of radius r , th...

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  17. Pa n dQ are two points on a circle of centre C and radius alpha . The ...

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  18. The lower corner of a leaf in a book is folded over so as to reach the...

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  19. Let A(p^2,-p),B(q^2, q),C(r^2,-r) be the vertices of triangle ABC. A p...

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  20. A window of perimeter P (including the base of the arch) is in the ...

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