Marks 1
Let $f(x)=\frac{e^x-e^{-x}}{2}, x \in R$. Let $f^{(k)}(a)$ denote the $k^{\text {th }}$ derivative of $f$ evaluated at $a$. What is the value of $f^{(10)}(0)$ ?(Note: ! denotes factorial)
Consider two functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow(1, \infty)$. Both functions are differentiable at a point c . Which of the following functions is/are ALWAYS differentiable at c ? The symbol $\cdot$ denotes product and the symbol odenotes composition of functions.
$$\mathop {\lim }\limits_{t \to + \infty } \sqrt{t^2+t}-t= $$
(Round
off to one decimal place)
Marks 2
Consider the function
$$ f(\mathrm{x})=\frac{x^3}{3}+\frac{7}{2} x^2+10 x+\frac{133}{2}, x \in[-8,0] . $$
Which of the following statements is/are correct?
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice-differentiable function and suppose its second derivative
satisfies $f^{\prime \prime}(x)>0$ for all $x \in \mathbb{R}$. Which of the following statements is/are ALWAYS correct?
Let $\quad f: \mathbb{R} \rightarrow \mathbb{R} \quad$ be such that $|f(x)-f(y)| \leq(x-y)^2$ for all $x, y \in \mathbb{R}$.
Then $\quad f(1)-f(0)=$ ____________