For the graph of the function y = f(x) , a line y = b is called an asymptote if for the distance d = \left| {f({x_i}) - b} \right|

• A) d \to b
• B) d \to 1
• C) d \to 0
• D) d \to \infty

If{\text{ }}{\left( {{f_{xy}}} \right)^2} - {f_{xx}}{f_{yy}} = {\text{ }}0,{\text{ }}then{\text{ }}the{\text{ }}double{\text{ }}point{\text{ }}\left( {x,{\text{ }}y} \right){\text{ }}would{\text{ }}be{\text{ }}a

• A) Cusp
• B) Isolated point
• C) None of above
• D) Node

If{\text{ }}f'\left( x \right) < 0{\text{ }}on{\text{ }}an{\text{ }}open{\text{ }}interval{\text{ }}extending{\text{ }}left{\text{ }}from{\text{ }}{x_0}\,and{\text{ }}f'\left( x \right) > 0{\text{ }}on{\text{ }}an{\text{ }}open{\text{ }}interval{\text{ }}extending{\text{ }}right{\text{ }}from{\text{ }}{x_0},{\text{ }}then{\text{ }}at{\text{ }}{x_0}\,the{\text{ }}f{\text{ }}has{\text{ }}a

• A) No maxima
• B) Relative maximum
• C) Relative extremum
• D) Relative minimum

If for a rational function r(x) = \frac{{p(x)}}{{q(x)}} , degree of p(x) is strictly one great than the degree of q(x) then r(x) will have a/an . . . . . . .

• A) Oblique asymptote
• B) y = 1 line as asymptote
• C) Horizontal asymptote
• D) Vertical asymptote

Which of the following cannot be an asymptote to the graph of a function y = f(x)

• A) Vertical Asymptote
• B) Curved Asymptote
• C) Oblique Asymptote
• D) Horizontal Asymptote

For the graph of a function y = f(x) , asymptote is a line that as x \to \infty, . . . . . . . .

• A) Crosses the curve after constant intervals.
• B) Gets arbitrarily away from the curve.
• C) Remains at a constant distance from the curve.
• D) Gets arbitrarily close to the curve but never touches it.

For a rational function r(x) = \frac{{p(x)}}{{q(x)}} = \frac{{{a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ... + {a_0}}}{{{b_m}{x^m} + {b_{m - 1}}{x^{m - 1}} + ... + {b_0}}} , if n > m, then . . . . . . . .

• A) The line y = {a_n} is the horizontal asymptote.
• B) The line y = {b_m} is the vertical asymptote.
• C) The rational function r(x) becomes unbounded for large values of x.
• D) The line y = \frac{{{a_n}}}{{{b_m}}} is the horizontal asymptote.

Which of the following is true for the function f(x) = {x^{2\,}} - 2{\text{ }};\,x \in \mathbb{R}

• A) It does not have an absolute minimum value
• B) It has an absolute maximum value at x = 0
• C) It has an absolute minimum value at x=0
• D) None of the others

The{\text{ }}critical{\text{ }}points{\text{ }}of{\text{ }}the{\text{ }}polynomial{\text{ }}p\left( x \right) = {x^3} - 2{x^2}\,are

• A) 0,-2
• B) ±2
• C) 2
• D) 0,4/3

Let{\text{ f(x) = - 10x + 50, }}and{\text{ }}\frac{{df(x)}}{{dx}} < 0,\,then

• A) f(x) must has a critical point
• B) f(x) is a decreasing function
• C) f(x) is an increasing function
• D) None of the other

According{\text{ }}to{\text{ }}the{\text{ }}second{\text{ }}derivative{\text{ }}test{\text{ }}f{\text{ }}has{\text{ }}a{\text{ }}relative{\text{ }}minimum{\text{ }}value{\text{ }}at{\text{ }}c{\text{ }}if{\text{ }}f''\left( c \right)

• A) ≥0
• B) >0
• C) <0
• D) =0

\[According{\text{ }}to{\text{ }}the{\text{ }}second{\text{ }}derivative{\text{ }}test{\text{ }}f{\text{ }}has{\text{ }}a{\text{ }}relative{\text{ }}minimum{\text{ }}value{\text{ }}at{\text{ }}c{\text{ }}if{\text{ }}f''\left( c \right)\]

• A) =0
• B) <0
• C) >0
• D) ≥0

\[A{\text{ }}point{\text{ }}on{\text{ }}the{\text{ }}curve{\text{ }}through{\text{ }}which{\text{ }}r{\text{ }}branches{\text{ }}of{\text{ }}the{\text{ }}curve{\text{ }}pass{\text{ }}is{\text{ }}called{\text{ }}Multiple{\text{ }}point{\text{ }}of\]

• A) Multiple order
• B) pth order
• C) rth order
• D) sth order

The{\text{ }}critical{\text{ }}points{\text{ }}of{\text{ }}the{\text{ }}polynomial{\text{ }}p\left( x \right) = {x^3} - 3x + 1{\text{ }}are

• A) 0
• B) ±1
• C) -1
• D) 1

If{\text{ }}f\left( x \right) = {\left( {x - 1} \right)^{2/3}} - 3\left( {x - 1} \right){\text{ }}and{\text{ }}f'\left( x \right) = \frac{2}{{3{{\left( {x - 1} \right)}^{1/3}}}} - 3{\text{ }}then{\text{ }}the{\text{ }}singular{\text{ }}point{\text{ }}of{\text{ }}f\left( x \right){\text{ }}is

• A) -1
• B) 2
• C) 0
• D) 1

Which of the following is true for the function f(x) = 5{x^2} - 15x - 20{\text{ }};x \in \mathbb{R}

• A) It has a critical number at x = 3/2
• B) It has a critical point at x= 20
• C) It has no critical point
• D) None of the other

If{\text{ }}f'\left( x \right){\text{ }}has{\text{ }}the{\text{ }}same{\text{ }}sign{\text{ }}on{\text{ }}both{\text{ }}left{\text{ }}and{\text{ }}right{\text{ }}sides{\text{ }}of{\text{ }}{x_0}\,on{\text{ }}an{\text{ }}open{\text{ }}interval,{\text{ }}then{\text{ }}f.........\,relative{\text{ }}extremum/extrema{\text{ }}at{\text{ }}{x_0}.

• A) Does not have
• B) One
• C) Three
• D) Two

\[If{\text{ }}{\left( {{f_{xy}}} \right)^2} - {f_{xx}}{f_{yy}} = {\text{ }}0,{\text{ }}then{\text{ }}the{\text{ }}double{\text{ }}point{\text{ }}\left( {x,{\text{ }}y} \right){\text{ }}would{\text{ }}be{\text{ }}a\]

• A) None of above
• B) Isolated point
• C) Node
• D) Cusp

If{\text{ }}\frac{{df}}{{dx}}\, > \,0,\,\;at\,R(x,y),then

• A) f(x) is an increasing function at R(x,y)
• B) f(x) must has a critical point at R(x,y)
• C) f(x) is a decreasing function at R(x,y)
• D) None of the other

Which of the following is true for the functionf(x) = 3{x^2} - 6x - 10{\text{ }};x \in \mathbb{R}

• A) It has a critical point at x = 1
• B) None of the other
• C) It has no critical point
• D) It has a critical number at x= -1