Advertisements
Advertisements
प्रश्न
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
उत्तर
\[\text{ Here }, \]
\[f\left( x \right) = \frac{1}{x}\]
\[\text { Let } x_1 , x_2 \in \left( 0, \infty \right) \text { such that } x_1 < x_2 . \text { Then }, \]
\[ x_1 < x_2 \]
\[ \Rightarrow \frac{1}{x_1} > \frac{1}{x_2}\]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So, }f\left( x \right)\text { is decreasing on }\left( 0, \infty \right) .\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 12x2 + 18x + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
Find `dy/dx,if e^x+e^y=e^(x-y)`
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure Ec of a person with income I is given as Ec = ( 0.0003 ) I2 + ( 0.075 ) I when I = 1000.
Show that f(x) = x – cos x is increasing for all x.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
(a) Strictly increasing
(b) strictly decreasing
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
For every value of x, the function f(x) = `1/7^x` is ______
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
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 x is ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
Which of the following graph represent the strictly increasing function.
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
The function f(x) = x3 + 3x is increasing in interval ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
