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## Curve Sketching

You don't need a calculator or computer to draw your graphs! Derivatives and other Calculus techniques give direct insights into the geometric behavior of curves.

# Level 3

Let $$f(x) = x^3 + ax^2 + bx + c$$, where $$a, b$$, and $$c$$ are real numbers. In order for $$f(x)$$ to be invertible, $$a$$ and $$b$$ must be related as: $$\dfrac{a^m}{b^n} \leq p$$, where $$m, n$$, and $$p$$ are also real numbers.

Find the mimimum value of $$m + n + p$$.

What is the acute angle between the curves $$y=\sin x$$ and $$y=\cos x$$?

Note: Angle between two curves is the angle between the tangent lines at the point of intersection of the two curves.

A tangent to the function $$f(x)=\dfrac{1}{x}$$ is drawn on the first quadrant of the Cartesian plane.

Let $$X$$ denote the area bounded by this tangent, the $$x$$-axis and the $$y$$-axis.

What is the range of the values of $$X$$ can take?

$6 \ln (x^2 + 1) - x = 0$

How many real solutions exist for the equation above?

Which of the given options does the graph represent?

$\quad \begin{matrix} (a)\quad \left| y \right| =\cos { x } \quad & (b)\quad \left| y \right| =\sin { x\quad \quad } \\ (c)\quad y=\left| \cos { x } \right| \quad & (d)\quad \left| y \right| =\left| \cos { x } \right| \end{matrix}$

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