10. If the function $f ( x ) = a | x - b | + 2$ is increasing on $[ 0 , + \infty )$, then the range of real numbers $a$ and $b$ is $\_\_\_\_$.

Let $f(x) = a_0 + a_1 |x| + a_2 |x|^2 + a_3 |x|^3$, where $a_0, a_1, a_2, a_3$ are constants. Then
(A) $f(x)$ is differentiable at $x = 0$ whatever be $a_0, a_1, a_2, a_3$
(B) $f(x)$ is not differentiable at $x = 0$ whatever be $a_0, a_1, a_2, a_3$
(C) $f(x)$ is differentiable at $x = 0$ only if $a_1 = 0$
(D) $f(x)$ is differentiable at $x = 0$ only if $a_1 = 0, a_3 = 0$

Let $f ( x ) = a _ { 0 } + a _ { 1 } | x | + a _ { 2 } | x | ^ { 2 } + a _ { 3 } | x | ^ { 3 }$, where $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ are constants. Then
(a) $f ( x )$ is differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(b) $f ( x )$ is not differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(c) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$
(d) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$.

Let $f ( x ) = a _ { 0 } + a _ { 1 } | x | + a _ { 2 } | x | ^ { 2 } + a _ { 3 } | x | ^ { 3 }$, where $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ are constants. Then
(a) $f ( x )$ is differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(b) $f ( x )$ is not differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(c) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$
(d) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$.

Let $f(x) = a_0 + a_1 |x| + a_2 |x|^2 + a_3 |x|^3$, where $a_0, a_1, a_2, a_3$ are constants.
(A) $f(x)$ is differentiable at $x = 0$ whatever be $a_0, a_1, a_2, a_3$
(B) $f(x)$ is not differentiable at $x = 0$ whatever be $a_0, a_1, a_2, a_3$ Then
(C) $f(x)$ is differentiable at $x = 0$ only if $a_1 = 0$
(D) $f(x)$ is differentiable at $x = 0$ only if $a_1 = 0, a_3 = 0$

Let $f ( x ) = a _ { 0 } + a _ { 1 } | x | + a _ { 2 } | x | ^ { 2 } + a _ { 3 } | x | ^ { 3 }$, where $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ are constants.
(A) $f ( x )$ is differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(B) $f ( x )$ is not differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ Then
(C) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$
(D) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$

Let $f(x) = a_0 + a_1|x| + a_2|x|^2 + a_3|x|^3$, where $a_0, a_1, a_2, a_3$ are constants. Then $f'(x)$ exists at $x = 0$ if and only if
(A) $a_1 = 0$
(B) $a_1 = 0$ and $a_2 = 0$
(C) $a_1 = 0$ and $a_3 = 0$
(D) $a_2 = 0$ and $a_3 = 0$

The number of points at which the function $$f(x) = |2x+1| - 3|x+2| + |x^2 + x - 2|, \quad x \in \mathbb{R}$$ is NOT differentiable is ____.

Let $f ( x ) = 15 - | x - 10 | ; x \in R$. Then the set of all values of $x$, at which the function $g ( x ) = f ( f ( x ) )$ is not differentiable, is:
(1) $\{ 5,10,15 \}$
(2) $\{ 10 \}$
(3) $\{ 10,15 \}$
(4) $\{ 5,10,15,20 \}$

Let $S$ be the set of points where the function, $f(x) = |2 - |x - 3||$, $x \in R$, is not differentiable. Then $\sum _ { x \in S } f(f(x))$ is equal to

The number of points, at which the function $f ( x ) = | 2 x + 1 | - 3 | x + 2 | + \left| x ^ { 2 } + x - 2 \right| , x \in R$ is not differentiable, is

Let $f(x) = |2x^2 + 5x - 3|$, $x \in \mathbb{R}$. If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m + n$ is equal to:
(1) 5
(2) 2
(3) 0
(4) 3

I. $f ( x ) = x - 1$ II. $g ( x ) = | x - 1 |$ III. $h ( x ) = \sqrt [ 3 ] { ( x - 1 ) ^ { 2 } }$ Which of the following functions do not have a derivative at the point $x = 1$?
A) Only I
B) Only II
C) I and II
D) II and III
E) I, II and III