Let a function $f: R \rightarrow R$ be defined as :
$f(x)=\begin{cases} \int_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x, & x \leq 4 \end{cases}$
where $b \in R$. If $f$ is continuous at $x=4$,then which of the following statements is $NOT$ true?

Consider the function $f(x) = x \cos x - \sin x$. Identify the correct statement.

Let $f(x)$ be a quadratic expression which is positive for all real $x$. If $g(x) = f(x) + f'(x) + f''(x)$,then for any real $x$,which one is correct?

Consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \begin{cases} (2 - \sin(\frac{1}{x}))|x|, & x \neq 0 \\ 0, & x = 0 \end{cases}$. Then $f$ is

If $\alpha$ is the angle of intersection between the curves $y = a^x$ and $y = b^x$,then what is $\tan \alpha$ equal to?

Give the correct order of initials $T$ or $F$ for the following statements. Use $T$ if the statement is true and $F$ if it is false.
Statement-$1$: If $f: R \rightarrow R$ and $c \in R$ is such that $f$ is increasing in $(c - \delta, c)$ and $f$ is decreasing in $(c, c + \delta)$,then $f$ has a local maximum at $c$. Where $\delta$ is a sufficiently small positive quantity.
Statement-$2$: Let $f: (a, b) \rightarrow R, c \in (a, b)$. Then $f$ cannot have both a local maximum and a point of inflection at $x = c$.
Statement-$3$: The function $f(x) = x^2 |x|$ is twice differentiable at $x = 0$.
Statement-$4$: Let $f: [c - 1, c + 1] \rightarrow [a, b]$ be a bijective map such that $f$ is differentiable at $c$ and $f'(c) \neq 0$,then $f^{-1}$ is also differentiable at $f(c)$.