Advertisements
Advertisements
प्रश्न
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
उत्तर
\[f\left( x \right) = \frac{1}{1 + x^2}\]
\[\text { R can be divided into two intervals }\left( 0, \infty \right)\text { and }( - \infty , 0] . \]
\[\text { Case }1: \text { Let } x_1 , x_2 \in \left( 0, \infty \right) \text { such that } x_1 < x_2 . \text { Then },\]
\[ x_1 < x_2 \]
\[ \Rightarrow {x_1}^2 < {x_2}^2 \]
\[ \Rightarrow 1 + {x_1}^2 < 1 + {x_2}^2 \]
\[ \Rightarrow \frac{1}{1 + {x_1}^2} > \frac{1}{1 + {x_2}^2}\]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right) \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So, }f\left( x \right) \text { is decreasing on }\left( 0, \infty \right).\]
\[\text { Case } 2: \text { Let } x_1 , x_2 \in ( - \infty , 0] \text { such that } x_1 < x_2 . \text { Then },\]
\[ x_1 < x_2 \]
\[ \Rightarrow {x_1}^2 > {x_2}^2 \]
\[ \Rightarrow 1 + {x_1}^2 > 1 + {x_2}^2 \]
\[ \Rightarrow \frac{1}{1 + {x_1}^2} < \frac{1}{1 + {x_2}^2}\]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right) \forall x_1 , x_2 \in ( - \infty , 0]\]
\[\text { So },f\left( x \right)\text { is increasing on }( - \infty , 0].\]
\[\text { Here }, f\left( x \right)\text { is decreasing on}\left( 0, \infty \right)\text { and increasing on }( - \infty , 0].\]
\[\text { Thus },f\left( x \right) \text { is neither increasing nor decreasing on R } . \]
APPEARS IN
संबंधित प्रश्न
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Show that f(x) = e2x is increasing on R.
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
The function f(x) = cot−1 x + x increases in the interval
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
Function f(x) = loga x is increasing on R, if
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
The total cost function for production of articles is given as C = 100 + 600x – 3x2, then the values of x for which the total cost is decreasing is ______
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
The function f(x) = sin x + 2x is ______
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
y = log x satisfies for x > 1, the inequality ______.
The function f(x) = x3 + 3x is increasing in interval ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
