Mathematics - Maxima and Minima
Exam Duration: 45 Mins Total Questions : 30
The points of extremum of the function \(F(x)=\int _{ 1 }^{ x }{ { e }^{ -{ t }^{ 2 }/2 } } (1-{ t }^{ 2 })\) dt are
- (a)
±1
- (b)
0
- (c)
±1/2
- (d)
±2
The difference between the greatest and the least value of the function \(f\left( x \right) =\int _{ 0 }^{ x }{ \left( { t }^{ 2 }+t+1 \right) } dt\) on [2,3] is
- (a)
37/6
- (b)
47/6
- (c)
57/6
- (d)
59/6
Let \(f(x)=\begin{cases} \left| { x }^{ 3 }+{ x }^{ 2 }+3x+sinx \right| \left( 3+sin\frac { 1 }{ x } \right) ,x\neq 0 \\ 0,\quad x=0 \end{cases}\)then number of points (where f(x) attains its minimum value) is
- (a)
1
- (b)
2
- (c)
3
- (d)
infinite many
A differentiable function f(x) has a relative minimum at x = 0, then the function y = f(x) + ax + b has a relative minimum at x = 0 for
- (a)
all a and all b
- (b)
all b if a = 0
- (c)
all b > 0
- (d)
all a > 0
N Characters of information are held on magnetic tape, in batches of x characters each, the batch processing time is ∝+βx2 seconds, ∝ and β are constants. The optical value of x for fast processing is
- (a)
\(\frac { \alpha }{ \beta } \)
- (b)
\(\frac { \beta }{ \alpha } \)
- (c)
\(\sqrt { \frac { \alpha }{ \beta } } \)
- (d)
\(\sqrt { \frac { \beta }{ \alpha } } \)
The minimum value of \(\left( 1+\frac { 1 }{ { sin }^{ n }\alpha } \right) \left( 1+\frac { 1 }{ { cos }^{ n }\alpha } \right) \) is
- (a)
1
- (b)
2
- (c)
(1+2n/2)2
- (d)
none of these
Let f(x) be a function such that f' (a)≠0. Then at x = a, f(x)
- (a)
cannot have a maximum
- (b)
cannot have a minimum
- (c)
must have neither a maximum nor a minimum
- (d)
none of the above
A cylindrical gas container is closed at the top and open at the bottom; if the iron plate of the top is 5/4 times as thick as the plate forming the cylindrical sides. The ratio of the radius to the height of the cylinder using minimum material for the same capacity is
- (a)
2/3
- (b)
1/2
- (c)
4/5
- (d)
1/3
Let \(f(x)=\begin{cases} { x }^{ 3 }-{ x }^{ 2 }+10x-5,\quad x\le 1 \\ -2x+{ lo }g_{ 2 }\left( { b }^{ 2 }-2 \right) ,\quad x>1 \end{cases}\)the set of values of b for which f(x) have greatest value at x = 1 is given by
- (a)
1≤b≤2
- (b)
b={1,2}
- (c)
b∈(-∞,-1)
- (d)
none of these
Let f:[a,b]⇾R be a function such that for c ∈ (a,b), f'(c)=f''(c)=f'''(c)=fiv(c)=fv(c)=0 then
- (a)
f has local extremum at x = c
- (b)
f has neither local maximum nor local minimum at x=c
- (c)
f is necessarily a constant function
- (d)
it is difficult to say whether (a) or (b)
The number of solutions of the equation a f(x)+g(x)=0, where a > 0, g(x)≠0 and has minimum value 1/2 is
- (a)
one
- (b)
two
- (c)
infinite many
- (d)
zero
Two towns A and Bare 60 km apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all 200 students is to be as small as possible, then the school should be built at
- (a)
town B
- (b)
45 km from town A
- (c)
town A
- (d)
45 km from town B
The critical points of the function f'(x) where f(x)=\(\frac { \left| x-2 \right| }{ { x }^{ 2 } } \)is
- (a)
0
- (b)
2
- (c)
4
- (d)
6
Let f(x)=\(\int _{ 0 }^{ x }{ \frac { cost }{ t } } dt\) (x>0); then f(x) has
- (a)
maxima, when n = - 2, - 4, - 6, ...
- (b)
maxima, when n = -1, - 3, - 5, ...
- (c)
minima, when n = 0, 2, 4, ...
- (d)
minima, when n = 1, 3, 5, ...
Four points A, B, C and D lie in that order on the parabola y = ax2+bx+c and the coordinates of A, Band D are known A(- 2, 3); B(- 1, 1); D(2, 7).
The value of a is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
Four points A, B, C and D lie in that order on the parabola y = ax2+bx+c and the coordinates of A, Band D are known A(- 2, 3); B(- 1, 1); D(2, 7).
The value of b is
- (a)
-3
- (b)
-1
- (c)
0
- (d)
1
Four points A, B, C and D lie in that order on the parabola y = ax2+bx+c and the coordinates of A, Band D are known A(- 2, 3); B(- 1, 1); D(2, 7).
The value of c is
- (a)
3
- (b)
2
- (c)
0
- (d)
none of these
A cubic f(x)=ax3+bx2+cx+d vanishes at x=-2 and has relative minimum/maximum at x = - 1 and x=\(\frac{1}{3}\) and if \(\int _{ -1 }^{ 1 }{ f(x)dx=\frac { 14 }{ 3 } } \)
The value of c is
- (a)
-2
- (b)
-1
- (c)
0
- (d)
2
A cubic f(x)=ax3+bx2+cx+d vanishes at x=-2 and has relative minimum/maximum at x = - 1 and x=\(\frac{1}{3}\) and if \(\int _{ -1 }^{ 1 }{ f(x)dx=\frac { 14 }{ 3 } } \)
The function f(x) is
- (a)
x3+x2+x-2
- (b)
x3-x2+x-2
- (c)
x3-x2-x-2
- (d)
x3+x2-x-2
A cubic f(x)=ax3+bx2+cx+d vanishes at x=-2 and has relative minimum/maximum at x = - 1 and x=\(\frac{1}{3}\) and if \(\int _{ -1 }^{ 1 }{ f(x)dx=\frac { 14 }{ 3 } } \)
The nature of roots of f(x) = 3 is
- (a)
one root is real and other two are distinct
- (b)
all roots real and distinct
- (c)
all roots are real; two of them are equal
- (d)
none of the above
The point (0,5) is closest to the curve x2=2y at
- (a)
(2√2,0)
- (b)
(2,2)
- (c)
(-2√2,0)
- (d)
(2√2,4)
The least value of 'a' for which \(\frac { 4 }{ sinx } +\frac { 1 }{ 1-sinx } \)=a has atleast one solution in the interval (0,π/2)
- (a)
9
- (b)
8
- (c)
4
- (d)
1
If f(x)=\(\begin{cases} 1+{ x }^{ 2 }-3x,\quad x<0 \\ cosx+2x,\quad x\ge 0 \end{cases}\) , then the global maximum and local minimum values of f(x) for x∈[-2,2] are respectively.
- (a)
4+cos 2,1
- (b)
11,1
- (c)
11, not exist
- (d)
none of these
The coordinate of the point on y2=8x, which is closest from x2+(y+6)2=1 is/are
- (a)
(2,-4)
- (b)
(18,-12)
- (c)
(2,4)
- (d)
none of these