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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.

# Curve Sketching: Level 4 Challenges

How many real values of $$x$$ satisfy the equation

$\large {x}^{2}-{2}^{x}=0?$

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

How many real solutions exist for the equation above?

On a vertical parabola $$P_1$$, pick a point $$T$$. Another vertical parabola $$P_2$$ is tangent to $$P_1$$ at $$T$$. Spanning through all possible parabolas $$P_2$$, what is the locus of the focus of $$P_2$$?

Let $$f(x)=x^4-6x^2+5.$$ If $$P(x_0,y_0)$$ is a point such that $$y_0>f(x_0)$$ and there are exactly two distinct tangents drawn to the curve $$y=f(x),$$ what is the maximum value of $$y_0?$$

Try : Part-2

The function $$f(x)= x^{3}-11x^{2} +19x+13$$ has zeros $$a_{1},a_{2},a_{3}.$$ Then what is the value of $$[ a_{1} ]+[ a_{2} ]+[ a_{3} ]?$$

Note: $$[m]$$ Represents the greatest integer less than or equal to $$m.$$

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