If f(x) = Cos(x) + Sin(x) + x, then which of the following is NOT true about it.

• A) Its anti – derivative is Sin(x) – Cos(x) + x^2/2 + 4.
• B) Its anti – derivative is Sin(x) – Cos(x) + x^2/2 + 5.
• C) None of these.
• D) Its anti – derivative is -Sin(x) – Cos(x) + x^2/2 + 4.

If a function has an extreme value (either a maximum or a minimum) on an open interval (a,b), then the extreme value occurs at a ........

• A) Any point in the open interval (a,b)
• B) Critical point of f(x)
• C)
• D)

f(x)=x^3-3x, is increasing on the interval (0 ,infinity)

• A) True
• B) False
• C)
• D)

If f (x) = x^2 is defined on the interval [-1, 3], then which of the following is true about it.

• A) Its absolute maximum value is 0.
• B) None of these.
• C) Its absolute maximum value is 9.
• D) Its relative maximum value is 9.

If the graph of $f$ lies below all of its tangents on an interval $I$,then it is called___________ on $I$.

• A) None of these
• B) concave downward
• C) concave upward
• D) constant function

$${\text{For}}\,{\text{Rolle's}}\,{\text{Theorem,}}\,f\,{\text{is}}\,{\text{continuous}}\,{\text{on}}\,{\text{the}}\,{\text{interval __________}}{\text{.}}$$

• A) $$[a,b)$$
• B) $$(a,b)$$
• C) $$(a,b]$$
• D) $$[a,b]$$

If the function f(x)= (x^2)+1 satisfies all the conditions of Rolle’s theorem in the interval [-1,1] , then the value of ‘c’ will be……..

• A) 2
• B) 3
• C) 4
• D) 0

What will be the sigma notation for 32+42+52+62 ?

• A) summation of (k+1)2 where (k varies from 3 to 6)
• B) summation of (k2) where (k varies from 3 to 6)
• C) summation of (k-1)2 where (k varies from 2 to 5)
• D) summation of (k2) where (k varies from 1 to 4)

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 )

The estimated area under f(x) = x^2 from
x = 1 to x = 3 with right end points for
n = 2 is ________.

• A) 10.
• B) 5.
• C) 13.
• D) None of these.

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

• A) 0, 6 and 12
• B) 4, 8 and 12
• C) 2, 7 and 12
• D) 6, 9 and 12

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

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

If f(x) = Sec^ (2) x + x^3, then which of the following is NOT true about it.

• A) None of these.
• B) Its anti – derivative is Tan(x) + x^4/4 + 15.
• C) Its anti – derivative is Tan(x) + x^4/4 + 12.
• D) Its anti – derivative is Tan(x) + x^4/4 + 10.

Area of a rectangle whose width is 5 units and length is 6 units will be ….

• A) 22 units
• B) 30 square units
• C) 11 units
• D) None of these

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

• A) \[{\text{ - }}\frac{{\cos 5x}}{5} + c\]
• B) \[\frac{{\cos 5x}}{5} + c\]
• C) \[5\cos 5x + c\]
• D) \[{\text{ - }}\frac{{\cos 4x}}{5} + c\]

Mean value theorem states that between any two points A and B on a curve y=f(x) , there must be at least one point where the Tangent line to the curve is……………joining A and B

• A) Parallel to the tangent line
• B) Perpendicular to the secant line
• C) Perpendicular to the tangent line
• D) Parallel to the secant line

By applying mean value theorem to the function
f(x)=lnx in the interval [1,e], the corresponding value of ‘c’ is ------------(Hint: lne=1, ln1=0)

• A) e-1
• B) -e
• C) e
• D) 1-e

$${\text{The integral }}\int {\sqrt {2x + 3} \,dx} {\text{ will be equal to ?}}$$

• A) $$\frac{{{{\left( {2x + 3} \right)}^{\frac{3}{2}}}}}{3} + c$$
• B) $$\frac{{{{\left( {2x + 3} \right)}^{\frac{1}{2}}}}}{3} + c$$
• C) $$\frac{{{{\left( {2x + 3} \right)}^{\frac{2}{3}}}}}{3} + c$$
• D) $$\frac{{{{\left( {2x + 3} \right)}^{\frac{3}{2}}}}}{2} + c$$

Summation of 2 where sum ranges from 0 to 10 equals 20.

• A) False
• B) True
• C)
• D)

If $f$ has a local maximum or minimum at c, and if $f'(c)$ exists, then $f'(c) = 0$. This is the statement of _________.

• A) Extreme Value Theorem
• B) Fermat's Theorem
• C) Mean Value Theorem
• D) None of these