### Mathematics - Differentiability and Differentiation

#### Question - 1

Consider the function f(x)=|logex|,$\forall$x>0.Then,

• A LHD does not exist at x=1
• B RHD does not exist at x=1
• C f is differentiable at x=1
• D f is not differential at x=1

#### Question - 2

Consider the function f(x) defined by $\\ f(x)=\begin{cases} xsin(ln{ x }^{ 2 }),x\neq 0 \\ 0,\quad \quad \quad \quad x=0 \end{cases}$.Then

• A f is differential at x=0
• B f is not differential at x=0
• C LHD exists but RHD does not exists at x=0
• D RHD exists but LHD does not exists at x=0

#### Question - 3

If $\sqrt{1-x^2}-\sqrt{1-y^2}=a(x-y)$, then$dx\over dy$is equal to

• A $\sqrt{1-x^2\over1-y^2}$
• B $\sqrt{1-y^2\over1-x^2}$
• C $\sqrt{x^2-1\over1-y^2}$
• D $\sqrt{y^2-1\over1-x^2}$

#### Question - 4

Let x=a(cost+logtan$t\over2$)and y=a sint, then $dy\over dx$is

• A cot t
• B tan t
• C -tan t
• D None of the above

#### Question - 5

Let f:[-5,5]$\rightarrow$R be a differentiable function such that f'(x) does not vanish anywhere, then

• A f(-5)>f(5)
• B f(-5)
• C f(-5)=f(5)
• D f(-5)$\neq$f(5)

#### Question - 6

If F(x)=f(x).g(x0 and f'(x).g'(x)=c, then

• A $F'=c[{f\over f'}+{g\over g'}]$
• B $F'=c[{f\over f'}-{g\over g'}]$
• C ${F''\over c}={f''\over f}+{g''\over g}+{2c\over fg}$

#### Question - 7

Observe the following columns

 Column I Column II A.The function    $f(x)=\begin{cases} { x }^{ 2 }+3x+a;\quad x\le 1 \\ bx+2;\quad \quad \quad x>1 \end{cases}$ differentiable,$\forall\ x\epsilon R$ then P.a=3 B.The function     $f(x)=\begin{cases} \frac { 1 }{ |x| } ;\quad \quad\quad |x|\le 1 \\ ax^{ 2 }+b;\quad |x|>1 \end{cases}$ differentiable everywhere, then Q.b=5 C.The function    $f(x)=\begin{cases} { ax }^{ 2 }-bx+2;\quad x<3 \\ bx^{ 2 }-3b;\quad \quad\quad x\ge 1 \end{cases}$ differentiable everywhere, then R.$a={35\over9}$ S.$b={3\over2}$ T.$a=-{1\over2}$
• A A B C PQ ST R
• B A B C R SR RP
• C A B C T SP RP
• D None of the above

#### Question - 8

Let f(x)=x2+xg'(1)+g''(2) and g(x)=x2+xf'(2)+f''(3), then

• A f'(1)=4-f'(2)
• B g'(2)=8-g'(1)
• C g''(2)+f''(3)=4
• D None of the above

#### Question - 9

Consider the function f(x) defined by$f(x)=\begin{cases} \frac { { x(e }^{ -1/x }-{ e }^{ 1/x } }{ { e }^{ -1/x }+{ e }^{ 1/x } } ,x\neq 0 \\ o,\quad \quad \quad \quad x=0 \end{cases}$.Then,

• A  f is continuous and derivable at x=0
• B f is continuous but not derivable at x=0
• C  f is not continuous at x=0
• D None of the above

#### Question - 10

Consider the function f(x) defined by f(x)=x-2|+|x|+|x+2|.Then,

• A f is derivable at x=0,2
• B f is derivable at x=-2,0
• C f is not derivable at x=-2,2
• D f is not derivable at x=-2,0,2