Let $h$ be a twice continuously differentiable positive function on an open interval $J.$ Let $g(x) = \ln(h(x))$ for each $x \in J$. Suppose $(h'(x))^2 > h''(x) h(x)$ for each $x \in J$. Then

Let $f: (-\infty, \infty) - \{0\} \rightarrow R$ be a differentiable function such that $f^{\prime}(1) = \lim_{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)$. Then $\lim_{a \rightarrow \infty} \left[ \frac{a(a+1)}{2} \tan^{-1}\left(\frac{1}{a}\right) + a^2 - 2 \log_e a \right]$ is equal to

Let $f(\theta)=3\left(\sin ^4\left(\frac{3 \pi}{2}-\theta\right)+\sin ^4(3 \pi+\theta)\right)-2\left(1-\sin ^2 2 \theta\right)$ and $S=\left\{\theta \in[0, \pi]: f^{\prime}(\theta)=-\frac{\sqrt{3}}{2}\right\}$. If $4 \beta=\sum_{\theta \in S} \theta$ then $f(\beta)$ is equal to

The number of points, at which the function $f(x) = \max\{6x, 2+3x^2\} + |x-1| |\cos(x^2 - 1/4)|, x \in (-\pi, \pi)$, is not differentiable, is ————

Let $f:R \to R$ be a continuously differentiable function such that $f(2) = 6$ and $f'(2) = \frac{1}{48}$. If $\int_6^{f(x)} 4t^3 \,dt = (x - 2)g(x)$,then $\lim_{x \to 2} g(x)$ is equal to

Let $f:[0,3] \rightarrow R$ be defined by $f(x)=\min \{x-[x], 1+[x]-x\}$,where $[x]$ is the greatest integer less than or equal to $x$. Let $P$ denote the set containing all $x \in[0,3]$ where $f$ is discontinuous,and $Q$ denote the set containing all $x \in(0,3)$ where $f$ is not differentiable. Then the sum of number of elements in $P$ and $Q$ is equal to $......$