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
Let f(x) = a -(x-3)8/9 then maxima of f(x) is
- (a)
3
- (b)
a-3
- (c)
a
- (d)
none of these
Let f(x) be a differential function for all x, if f(1) = - 2 and f' (x) ≥2 for all x in [1, 6], then minimum value of f(6) is equal to
- (a)
2
- (b)
4
- (c)
6
- (d)
8
The point in the interval [0,2π], where f(x)=ex sin x has maximum slope is
- (a)
\(\frac { \pi }{ 4 } \)
- (b)
\(\frac { \pi }{ 2 } \)
- (c)
\(\pi \)
- (d)
none of these
Let f(x) = 1 + 2x2 + 22 x4 + .... + 210 x20 Then f(x) has
- (a)
more than one minimum
- (b)
exactly one minimum
- (c)
at least one maximum
- (d)
none of the above
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)
\(\sqrt { \frac { \beta }{ \alpha } } \)
On [1, e], the least and greatest values of f(x) = x2 ln x is
- (a)
e,1
- (b)
1,e
- (c)
0,e2
- (d)
none of these
The fuel charges for running a train are proportional to the square of the speed generated in mile/h and costs Rs 48 per h at 16 miles/h. The most economical speed if the fixed charges ie, salaries etc amount to Rs 300 per h
- (a)
10 mile/h.
- (b)
20 mile/h
- (c)
30 mile/h
- (d)
40 rnile/h
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
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
Let f(x) = cos x sin 2x, then
- (a)
\(min\quad \begin{matrix} f(x) \\ x\in \left( -\pi ,\pi \right) \end{matrix}>-7/9\)
- (b)
\(min\quad \begin{matrix} f(x) \\ x\in \left( -\pi ,\pi \right) \end{matrix}>-9/7\)
- (c)
\(min\quad \begin{matrix} f(x) \\ x\in \left( -\pi ,\pi \right) \end{matrix}>-1/9\)
- (d)
\(min\quad \begin{matrix} f(x) \\ x\in \left( -\pi ,\pi \right) \end{matrix}>-2/9\)
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, ...
The function f(x) = 3 + 2 (a + 1) x + (a2 + 1) x2- x3 has a local minimum at x = x1 and local maximum at x = x2 such that x1<2
- (a)
\(\left( -\infty ,-\frac { 3 }{ 2 } \right) \)
- (b)
\(\left( -\frac { 3 }{ 2 } ,1 \right) \)
- (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
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).
If area of quadrilateral ABCD is greatest, then the coordinates of Care
- (a)
\(\left( \frac { 7 }{ 4 } ,\frac { 1 }{ 2 } \right) \)
- (b)
\(\left( \frac { 1 }{ 2 } ,\frac { 7 }{ 4 } \right) \)
- (c)
\(\left( \frac { 1 }{ 2 } ,-\frac { 7 }{ 4 } \right) \)
- (d)
\(\left(- \frac { 1 }{ 2 } ,\frac { 7 }{ 4 } \right) \)
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 d is
- (a)
5
- (b)
2
- (c)
0
- (d)
-4
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 \(\int _{ -1 }^{ 1 }{ f(x)dx=\frac { 14 }{ 3 } } \)
f(x) decreases in the interval
- (a)
\(\left( -\frac { 1 }{ 3 } ,1 \right) \)
- (b)
\(\left( -\frac { 1 }{ 3 } ,-1 \right) \)
- (c)
\(\left( -1,\frac { 1 }{ 3 } \right) \)
- (d)
\(\left( 1\frac { 3 }{ 2 } \right) \)
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 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
The difference between the greatest and the least values of the function \(f(x)=\int _{ 0 }^{ x }{ \left( { at }^{ 2 }+1+cos\quad t \right) dt, } \)a>0 for x∈ [2,3] is
- (a)
\(\frac { 19 }{ 3 } a+1+(sin3-sin2)\)
- (b)
\(\frac { 18 }{ 3 } a+1+2\quad sin3\)
- (c)
\(\frac { 18 }{ 3 } a-1+2\quad sin3\)
- (d)
none of these