$\text{A critical point for a function }f\text{ is any value of }x\text{ in the domain of }f\text{ at which}$

• A) \[{f}'\left( x \right)\le 0\]
• B) \[{f}'\left( x \right)>0\]
• C) \[f\text{ is differentiable }\]
• D) \[{f}'\left( x \right)=0\,\text{or }f\text{ is not differentiable}\]

If $f'(c) = 0$ and $f''(c) > 0$, then $f$ has a/an____________ at c.

• A) inflection point
• B) None of these
• C) local minimum
• D) local maximum

x=1 is a critical value of the function.
f(x)=(x-1)^3

NOTE: x^n means ‘x’ to the power ‘n’

• A) True
• B) False
• C)
• D)

To find the area between continuous curve f(x) and the closed interval [a,b] on x-axis, we take
-------------- of f(x) on the interval [a,b].

• A) derivative
• B) left hand limit
• C) right hand limit
• D) integral

summation of (ai) (i varies from 1 to n) , summation of (aj) (j varies from 1 to n),summation of (ak) (k varies from 1 to n)
All these three represents same summation.

• A) False
• B) True
• C)
• D)

What is the estimated area under f(x) = 12 / x from x = 1 to x = 3 with mid points for n = 2?

• A) 13.1
• B) 10.5
• C) 12.8
• D) 18.0

A function $f$ has an/a___________ at $c$ if $f(c) \leqslant f(x)\,\forall \,x \in \,D$, where $D$ is the domain of $f$.

• A) global minimum
• B) local minimum
• C) absolute minimum/global minimum
• D) absolute minimum

According to the Newton’s method, the equation Cosx=x (where x is in radian) has a solution between ……………

• A) Solution not exist
• B) 1 and -1
• C) 0 and -1
• D) 0 and 1

Integral of x^101 is

NOTE: x^n means ‘x’ to the power ‘n’

• A) None of these
• B) 101x^100 +C
• C) 102x^102+C
• D) (1/102)x^102+C

$$\text{If a function }f\text{ has a relative extrema at }{{x}_{0}}\text{, then}$$

• A) $${f}'\left( {{x}_{0}} \right)>0$$
• B) $$\text{either}\,{f}'\left( {{x}_{0}} \right)=0\,\text{or }f\text{ is not differentiable at }{{x}_{0}}$$
• C) $$f\text{ is differentiable at }{{x}_{0}}$$
• D) $${f}'\left( {{x}_{0}} \right)\le 0$$

In sigma notation 12+14+16+18+20 can be written as……….

• A) summation of (2k) where (k varies from 1 to5)
• B) summation of (k^2) where (k varies from 6 to10)
• C) summation of (2k) where (k varies from 6 to10)
• D) summation of (k) where (k varies from 6 to10)

If ‘n’ goes from 1 to any large EVEN number then the summation of ‘(-1)^n’ = ---------

• A) that specific large EVEN number
• B) 1
• C) -1
• D) zero

What will be the value of summation of k3 where k goes from 9 to 10

• A) 1998
• B) 1081
• C) 1100
• D) 1729

For the area under the curve f(x) = x+1 from x = 1 to x = 5 with mid points approximations for n = 2, what will be the values of xk* ?

• A) 1 and 3
• B) 2 and 4
• C) 3 and 4
• D) 2.5 and 3.5

The derivative of the area under the continuous function f(x)= 2+3Sinx in the interval[-pi,pi] is---------

• A) 2+3Sinx
• B) 2-3Sinx
• C) 2-3Cosx
• D) 2+3Cosx

If a function f has a relative extrema at a point c, then c is a critical point for f .

• A) False
• B) True
• C)
• D)

For the area under the curve f(x) = 2x from x = 0 to x = 12 with mid points approximations for n = 3, what will be the values of xk* ?

• A) 3, 7 and 11
• B) 2, 6 and10
• C) 2, 5 and 8
• D) 1, 5 and 9

\[{\text{The integral }}\int {\cos (5x)\,dx\,} {\text{will be equal to ?}}\]

• A) \[\frac{{\sin (5x)}}{5} + c\]
• B) \[5\sin (5x) + c\]
• C) \[{\text{None of these}}\]
• D) \[ - \frac{{\sin (5x)}}{5} + c\]

\[\text{If a function }f\text{ has a relative extrema at }{{x}_{0}}\text{, then}\]

• A) \[\text{either}\,{f}'\left( {{x}_{0}} \right)=0\,\text{or }f\text{ is not differentiable at }{{x}_{0}}\]
• B) \[{f}'\left( {{x}_{0}} \right)\le 0\]
• C) \[f\text{ is differentiable at }{{x}_{0}}\]
• D) \[{f}'\left( {{x}_{0}} \right)>0\]

For the application of mean value theorem on f( x ) =x^3-3x^2-2x ; [0,2], which of the following is true?

• A) f( x ) is continuous over [ 0, 2] and f(x) is differentiable over [ 0,2 )
• B) f( x ) is continuous over [ 0, 2) and f(x) is differentiable over ( 0,2 )
• C) f( x ) is continuous over [ 0, 2] and f(x) is not differentiable over ( 0,2 )
• D) f( x ) is continuous over [ 0, 2] and f(x) is differentiable over ( 0,2 )