Curve Sketching

Curve Sketching: Level 3 Challenges


The above graph shows the curves of x2x^2 and x3x^3. Clearly, the line y=0y=0 is tangent to both the curves. But there exists another such straight line which is tangent to both the curves. Find the equation of the line.

If the line can be represented in the form ax+by+c=0ax+by+c=0, such that a,b|a|,|b| and c|c| are positive integers and gcd(a,b,c)=1\gcd(|a|,|b|,|c|) = 1, enter your answer as a+b+c|a|+|b|+|c|.

Notation: | \cdot | denotes the absolute value function.

If f(x)=x{x}f'(x)=|x|-\{ x \}, then in which interval is the function f(x)f(x) decreasing?


{x}\{ x \} represents sawtooth function. That is, {x}=xx\{x\} = x - \lfloor x \rfloor

Which of the given options does the graph represent?

Notation: | \cdot | denotes the absolute value function.

Let f(x)=x3+ax2+bx+cf(x) = x^3 + ax^2 + bx + c, where a,ba, b, and cc are real numbers. In order for f(x)f(x) to be invertible, aa and bb must be related as: ambnp\dfrac{a^m}{b^n} \leq p , where m,nm, n, and pp are also real numbers.

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

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

How many real solutions exist for the equation above?


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