Let a function g:[0,4]→R be defined as
g(x)={\begin{matrix}\underset{0\leq t\leq x}{\mathrm{max}{{t}^{3}-6{t}^{2}+9t-3}}, & 0\leq x\leq 3 \\ 4-x, & 3<x\leq 4\end{matrix}
then the number of points in the interval (0,4) where g(x) is NOT differentiable, is _________.
Held on 20 Jul 2021 · Verified 6 Jul 2026.
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Let $y = y(x)$ be the solution of the differential equation $x\sin\left(\dfrac{y}{x}\right)dy = \left(y\sin\left(\dfrac{y}{x}\right) - x\right)dx$, $y(1) = \dfrac{\pi}{2}$ and let $\alpha = \cos\left(\dfrac{y(e^{12})}{e^{12}}\right)$. Then the number of integral values of $p$, for which the equation $x^2 + y^2 - 2px + 2py + \alpha + 2 = 0$ represents a circle of radius $r \leq 6$, is __________.
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