Mathematics - Maxima And Minima

Question - 1

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

Question - 2

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

Question - 3

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

Question - 4

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

Question - 5

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

Question - 6

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

Question - 7

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

Question - 8

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

Question - 9

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

Question - 10

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
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