JEE Main Mathematics - Maxima and Minima
Exam Duration: 60 Mins Total Questions : 30
A function f such that f' (2) = f" (2) = 0 and f has a local maximum of -17 at 2 is
- (a)
(x-2)4
- (b)
3-(x-2)4
- (c)
-17-(x-2)4
- (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
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
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)
\(\sqrt { \frac { \beta }{ \alpha } } \)
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 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
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
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,∞)
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).
If roots of ax2+bx+c=0 are ∝ and β, then equation whose roots are ∝19 and β7 is
- (a)
x2-x+1=0
- (b)
x2-2x+3=0
- (c)
x2+x+1=0
- (d)
x2+2x+3=0
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
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 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