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Differentiability

Those friendly functions that don't contain breaks, bends or cusps are "differentiable". Take their derivative, or just infer some facts about them from the Mean Value Theorem.

Level 3

         

\[\large{x^3+x^2-5x-1=0}\]

Let \(\alpha,\beta,\gamma\) be the roots of the equation above.

Evaluate \(\left\lfloor \alpha \right\rfloor +\left\lfloor \beta \right\rfloor +\left\lfloor \gamma \right\rfloor \).

Image Credit: Wikimedia N.Mori.

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

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

Find the equation of common tangent to the curves \(y^2=8x\) and \(xy=-1\)?

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

If minimum positive value of \(a\) can be expressed as \(\frac { A }{B }\) for co-prime \(A\) and \(B\), then find \(A+B\).

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