If first derivative is equal to zero, then these points are called ------------ .

• A) Double points
• B) Multiple points
• C) Critical points
• D) None of the other

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

• A) 0
• B) ±1
• C) 1
• D) -1

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

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

$$In{\text{ }}curve{\text{ }}{y^2} = x{\left( {x - a} \right)^2},{\text{ }}the{\text{ }}singular{\text{ }}point{\text{ }}\left( {a,0} \right){\text{ }}is{\text{ }}a{\text{ }} \ldots \ldots \,when{\text{ }}{f_{xx}}\left( {a,0} \right) = {\text{ }} - 2a,{\text{ }}{f_{yy}}\left( {a,0} \right) = 2{\text{ }}and{\text{ }}{f_{xy}}\left( {a,0} \right) = 0.$$

• A) Conjugate point
• B) Node
• C) Cusp
• D) Isolated point

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

• A) <0
• B) >0
• C) ≤0
• D) =0

The singular points of ${x^2}{(x - 3)^2}y'' + (x - 3)y' + 3{x^2}y = \,0$ are

• A) 1,3
• B) 4,3
• C) 0,3
• D) None of the other

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

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

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

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

Points of inflection and multiple points are the types of

• A) Singular point
• B) Double point
• C) None of these
• D) End points

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

• A) 0,4/3
• B) 2
• C) 0,-2
• D) ±2

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 point at x= 20
• B) It has no critical point
• C) It has a critical number at x = 3/2
• 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

A critical number for a function “f” . . . . . . . . exist(s) in the domain of function.

• A) Always
• B) May or may not
• C) None of the above
• D) Does not

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

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

A function f has a local minimum at a point “d” if ……………….. in some open interval containing “d”.

• A) $f(d)f(x)\, = 0$
• B) $f(d) < f(x)\,$
• C) $f(d)f(x)\, \geqslant 0\,\,$
• D) $f(d) \geqslant f(x)$

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

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

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

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

To discuss the nature of a double point, we have to calculate the

• A) Normal
• B) Both (a) and (b)
• C) Binormal
• D) Tangents

A double point Q on a curve is a ……… if there exist no real points of the curve in the neighborhood of R.

• A) Conjugate point
• B) Both (b) and (c)
• C) Complex point
• D) Isolated point

$$For{\text{ }}the{\text{ }}curve{\text{ }}{x^3} + {y^3} - 3axy = 0,{\text{ }}the{\text{ }}tangents{\text{ }}at{\text{ }}the{\text{ }}origin{\text{ }}are{\text{ }}x = 0{\text{ }}and{\text{ }}y = 0,{\text{ }}then{\text{ }}the{\text{ }}origin{\text{ }}is{\text{ }}a$$

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