Advertisements
Advertisements
प्रश्न
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
Advertisements
उत्तर
\[\text { Let } x_1 , x_2 \in \left( 0, \infty \right)\text { such that } x_1 < x_2 . \text { Then },\]
\[ x_1 < x_2 \]
\[ \Rightarrow \log_e x_1 < \log_e x_2 \]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So,}f\left( x \right) \text { is increasing on }\left( 0, \infty \right).\]
APPEARS IN
संबंधित प्रश्न
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
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) = 5x3 − 15x2 − 120x + 3 ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
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) ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
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 the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
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
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly 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 function f(x) = 9 - x5 - x7 is decreasing for
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The function f(x) = tan-1 x is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
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.
Let f(x) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.
