Advertisements
Advertisements
प्रश्न
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
Advertisements
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
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
The function f(x) = cot−1 x + x increases in the interval
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
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Show that f(x) = x – cos x is increasing for all x.
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Let 'a' be a real number such that the function f(x) = ax2 + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax2 – 6x + 15, x∈R has a ______.
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
If f(x) = x + cosx – a then ______.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
The function f(x) = xex(1 − x), x ∈ R, is ______.
As one moves from left to right on a graph, what does a decrease in the \[y\]-values indicate?
For \[f(x)=4x^3-6x^2-72x+30\], which factorization of \[f'(x)\] is correct?
For \[f'(x)=12(x-3)(x+2)\], what are the critical points obtained from \[f'(x)=0\]?
