IISER 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
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 \(f(x)=\begin{cases} \left| { x }^{ 3 }+{ x }^{ 2 }+3x+sinx \right| \left( 3+sin\frac { 1 }{ x } \right) ,x\neq 0 \\ 0,\quad x=0 \end{cases}\)then number of points (where f(x) attains its minimum value) is
- (a)
1
- (b)
2
- (c)
3
- (d)
infinite many
If \(f(x)={ alog }_{ e }\left| x \right| +{ bx }^{ 2 }+x\) has extremum at x = 1 and x = 3, then
- (a)
a = - 3/4, b = -1/8
- (b)
a = 3/4, b = -1/8
- (c)
a = - 3/4, b = 1/8
- (d)
none of these
Let \(f(x)=\begin{cases} { sin }^{ -1 }\alpha +{ x }^{ 2 },\quad 0<x<1 \\ 2x,\quad x\ge 1 \end{cases}\)f(x) can have a minimum at x = 1 is the value of ∝ is
- (a)
1
- (b)
-1
- (c)
0
- (d)
none of these
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
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:[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
If the function f(x)=2x3-9ax2+12a2x+1 has a local maximum at x=x1 and a local minimum at x=x2 such that x2=x12 then a is equal to
- (a)
0
- (b)
\(\frac{1}{4}\)
- (c)
2
- (d)
either (a) or (c)
Let \(\begin{cases} { x }^{ 2 }+3x,\quad -1\le x<0 \\ -sin\quad x,\quad 0\le x<\pi /2 \\ -1-cosx,\quad \frac { \pi }{ 2 } \le x\le \pi \end{cases}\) Then global maxima of f(x) and global minima of f(x) are equals
- (a)
-1
- (b)
0
- (c)
-3
- (d)
-2
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)=(x2-1)n (x2+x-1), then f(x) has local extremum at x = 1 when
- (a)
n=2
- (b)
n=3
- (c)
n=4
- (d)
n=6
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
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,∞)
If composite function f1(f2(f3(...(fn(x))...) is an increasing function and if r of fi' s are decreasing functions while rest are increasing, then maximum value of r (n - r) is
- (a)
\(\frac { { n }^{ 2 } }{ 4 } \), when n is an even number
- (b)
\(\frac { { n }^{ 2 } }{ 4 } \) when n is an odd number
- (c)
\(\frac { { n }^{ 2 }-1 }{ 4 } \)when n is an odd number
- (d)
\(\frac { { n }^{ 2 }-1 }{ 4 } \) when n is an even number
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 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 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 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)
\(\left( 1\frac { 3 }{ 2 } \right) \)
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 global maxima of f(x)=[2{-x2+x+1}] is (where {x} denotes fractional part of x and [.] denotes greatest integer function)
- (a)
2
- (b)
1
- (c)
0
- (d)
none of these
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 greatest value of the function f(x)=2 sin x+ sin 2x on the interval \(\left[ 0,\frac { 3\pi }{ 2 } \right] \) is
- (a)
\(\frac { 3\sqrt { 3 } }{ 2 } \)
- (b)
3
- (c)
\(\frac{3}{2}\)
- (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
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