Calculus

Differentiability

Differentiability: Level 3 Challenges

         

x3+x25x1=0\large{x^3+x^2-5x-1=0}

Let α,β,γ\alpha,\beta,\gamma be the roots of the equation above.

Evaluate α+β+γ\left\lfloor \alpha \right\rfloor +\left\lfloor \beta \right\rfloor +\left\lfloor \gamma \right\rfloor .

Image Credit: Wikimedia N.Mori.

Let ff be a twice differentiable function satisfying f(1)=1,f(2)=4,f(3)=9f(1)=1, f(2)=4, f(3)=9, then :

Let 0<a<b<π20<a<b<\frac { \pi }{ 2 } . If f(x)=sinxsinasinbcosxcosacosbtanxtanatanbf\left( x \right) =\left| \begin{matrix} \sin { x } & \sin { a } & \sin { b } \\ \cos { x } & \cos { a } & \cos { b } \\ \tan { x } & \tan { a } & \tan { b } \end{matrix} \right| , then minimum possible number of roots of f(x)=0f'\left( x \right) =0 lying in (a,b)(a,b) is:

Find the equation of common tangent to the curves y2=8xy^2=8x and xy=1xy=-1?

The curve of sin(ax)\sin { \left( ax \right) } is tangent to the curve of sin(x)\sin { \left( x \right) } at x=5π2x=\frac { 5\pi }{2 } .

If minimum positive value of aa can be expressed as AB\frac { A }{B } for co-prime AA and BB, then find A+BA+B.

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