Applicartions Of Derivatives Test

Applicartions Of Derivatives

This is Applicartions of Derivatives Test-04 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
The smallest value of the polynomial \({x^3} - 18{x^2} + 96\) x in the interval [0, 9] is
  • A
    0
    Correct
  • B
    135
  • C
    160
  • D
    126
2
Given that f (x) = \({x^{1/x}}\) , x\( > 0\;,\) has the maximum value at x = e,then
  • A
    \({e^\pi } < {\pi ^e}\)
  • B
    \({e^\pi } \leqslant {\pi ^e}\)
  • C
    \({e^\pi } = {\pi ^e}\)
  • D
    \({e^\pi } > {\pi ^e}\)
    Correct
3
If f (x) = \(x + \frac{1}{x},\) then
  • A
    relative minimum does not exist
  • B
    relative maximum > relative minimum
  • C
    relative minimum > relative maximum.
    Correct
  • D
    relative maximum does not exist
4
Let \(f(x) = {x^{25}}{(1 - x)^{75}}\) for all \(x \in \left[ {0,1} \right],\) then f (x) assumes its maximum value at
  • A
    \(\frac{1}{2}\)
  • B
    \(\frac{1}{3}\)
  • C
    0
  • D
    \(\frac{1}{4}\)
    Correct
5
Minimum value of the function\(f(x) = {x^2} + x + 1\) is
  • A
    none of these
  • B
    3
  • C
    $f(x) = {x^2} + x + 1$ $= {\left( {x + \frac{1}{2}} \right)^2} + \frac{3}{4} \geqslant \frac{3}{4}\forall x \in R$ $\frac{3}{4}$
    Correct
  • D
    1
6
If x be real, the minimum value of\({x^2} - 8x + 17\) is
  • A
    1
    Correct
  • B
    2
  • C
    – 1
  • D
    0
7
a log | x | + \(b\;{x^2}\) + x has its extreme values at x = – 1 and x = 2, then
  • A
    \(a = - 2,b = - \frac{1}{2}\)
  • B
    \(a = 2,b = - \frac{1}{2}\)
    Correct
  • C
    \(a = - 2,b = \frac{1}{2}\)
  • D
    a = 2, b = – 1
8
The maximum value of \(\frac{{{\text{log\;}}x}}{x}\) in \(0{\text{ }} < {\text{ }}x{\text{ }} < \infty \) is
  • A
    none of these
  • B
    – e
  • C
    \(\frac{1}{e}\)
    Correct
  • D
    0
9
f (x) = 1 + [ cos x ] x, in \(0 < x \leqslant \frac{\pi }{2}\)
  • A
    is continuous in \(\left( {0,\frac{\pi }{2}} \right]\)
    Correct
  • B
    has a maximum value 2
  • C
    has a minimum value 0
  • D
    is not differentiable at \(x = \frac{\pi }{3}\)
10
The function f (x) = \(x + \frac{4}{x}\) has
  • A
    local minima at x = 2 and a local maxima at x = – 2
    Correct
  • B
    a local maxima at x =2 and a local minima at x = – 2
  • C
    absolute minima at x = 2 and absolute maxima at x = – 2.
  • D
    absolute maxima at x = 2 and absolute minima at x = – 2
11
The maximum value of \(\frac{x}{{\log x}},x > 1,\) is
  • A
    \(\frac{1}{e}\)
  • B
    none of these
  • C
    e
    Correct
  • D
    – e
12
Maximum value of x –sin x in [0, π] is
  • A
    \(\frac{\pi }{2}\)
  • B
    \(\pi \)
    Correct
  • C
    none of these
  • D
    1
13
Minimum value of x + cos x in [0, π] is
  • A
    – 1 + π
  • B
    none of these
  • C
    1
    Correct
  • D
    \(\frac{\pi }{2}\)
14
Let f (x) =\({x^3}\), then f (x) has a
  • A
    local maxima at x = 0
  • B
    local minima at x = 0
  • C
    point of inflexion at x = 0
    Correct
  • D
    none of these
15
Let f ( x ) = \(x\left| x \right|\) , then f ( x )has
  • A
    cal minima at x = 0
  • B
    point of inflexion at x = 0
    Correct
  • C
    one of these
  • D
    local maxima at x = 0