Calculus

# Differentiability: Level 3 Challenges

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

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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