Let f be a real-valued function satisfying f(x+y)=f(x)+f(y) for all x, y. If f(1)=1/2, then the value of f(16) is: A. 16 B. 8 C. 4 D. 2

Let $f$ be a real-valued function satisfying $f\left(\frac{x}{y}\right) = f(x) - f(y)$ and $\lim_{{x \to 0}} \frac{f(1+x)}{x} = 3$. The area bounded by the curve $y=f(x)$, the $y$-axis, and the line $y=3$ is

• $9 e$
• $2 e$
• $3 e$
• None

Let $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ be a differentiable function satisfying $f(xy)=\frac{f(x)}{y}+\frac{f(y)}{x}$ for all $x, y \in \mathbb{R}^{+}$. Also $f(1)=0$ and $f'(1)=1$. If $M$ is the greatest value of $f(x)$, then $[m+e]$ is where $[.]$ represents the Greatest Integer Function.

A 3 B 2 C 0 D 1

Let $y = f(x)$ be a real-valued differentiable function on the set of all real numbers $\mathbb{R}$ such that $f(1) = 1$. If $f(x)$ satisfies $xf^{\prime}(x) = x^2 + f(x) - 2$, then

A. $f(x)$ is an even function.

B. $f(x)$ is an odd function.

C. The minimum value of $f(x)$ is 0.

D. $y = f(x)$ represents a parabola with focus $(1, \frac{5}{4})$.

Let the function f(x) = √(x^2 - |x + 2| + x) be a real valued function, then the domain of f(x) is:

• A [ (-∞, -√2] ∪ [√2, ∞) ]
• B ℝ
• C [ [-2, ∞) ]
• D ℝ - [ -√2, √2 ]

Let $f(x)=\left{\begin{array}{cl}\tan^{-1} x: & |x| \geq 1 \ \frac{x^{2}-1}{4}: & |x|<1 .\end{array}\right.$ Then domain of $f'(x)$ is:

(A) $\mathbb{R}-{1}$ (B) $\mathbb{R}-{-1,0,1}$

(C) $\mathbb{R}-{-1,1}$ (D) ${-1}$