Advertisements
Advertisements
प्रश्न
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
Advertisements
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
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 interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
Show that f(x) = e2x is increasing on R.
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
Every invertible function is
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
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
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.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
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 ______.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
A function f is said to be increasing at a point c if ______.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
