Curve Sketching
The above graph shows the curves of \(x^2\) and \(x^3\). Clearly, the line \(y=0\) 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 \(ax+by+c=0\), such that \(|a|,|b|\) and \(|c|\) are positive integers and \(\gcd(|a|,|b|,|c|) = 1\), enter your answer as \(|a|+|b|+|c|\).
\[\] Notation: \( | \cdot | \) denotes the absolute value function.
by
Arkajyoti Banerjee
If \(f'(x)=|x|-\{ x \}\), then in which interval is the function \(f(x)\) decreasing?
Details:
\(\{ x \}\) represents sawtooth function. That is, \(\{x\} = x - \lfloor x \rfloor\)
by
Sandeep Bhardwaj
Which of the given options does the graph represent?
\[\quad \begin{matrix} (a)\quad \left| y \right| =\cos { x } \quad & (b)\quad \left| y \right| =\sin { x\quad \quad } \\ (c)\quad y=\left| \cos { x } \right| \quad & (d)\quad \left| y \right| =\left| \cos { x } \right| \end{matrix}\]
by
Rohit Ner
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 \(m + n + p\).
by
M M
\[6 \ln (x^2 + 1) - x = 0 \]
How many real solutions exist for the equation above?
by
Sandeep Bhardwaj