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 intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Find the intervals in which the function f given by `f(x) = (4sin x - 2x - x cos x)/(2 + cos x)` is (i) increasing (ii) 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.
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) = 8 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
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 the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Function f(x) = ax is increasing on R, if
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
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.
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 the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
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 ______
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Show that function f(x) = tan x is increasing in `(0, π/2)`.
