Curve Sketching

Curve Sketching: Level 4 Challenges


How many real values of xx satisfy the equation

x22x=0?\large {x}^{2}-{2}^{x}=0?

6ln(x2+1)x=06 \ln (x^2 + 1) - x = 0

How many real solutions exist for the equation above?

On a vertical parabola P1P_1, pick a point TT. Another vertical parabola P2P_2 is tangent to P1P_1 at TT. Spanning through all possible parabolas P2P_2, what is the locus of the focus of P2P_2?

Let f(x)=x46x2+5.f(x)=x^4-6x^2+5. If P(x0,y0)P(x_0,y_0) is a point such that y0>f(x0)y_0>f(x_0) and there are exactly two distinct tangents drawn to the curve y=f(x),y=f(x), what is the maximum value of y0?y_0?

Try : Part-2

The function f(x)=x311x2+19x+13f(x)= x^{3}-11x^{2} +19x+13 has zeros a1,a2,a3.a_{1},a_{2},a_{3}. Then what is the value of [a1]+[a2]+[a3]?[ a_{1} ]+[ a_{2} ]+[ a_{3} ]?

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


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