Advertisements
Advertisements
प्रश्न
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Advertisements
उत्तर
\[f\left( x \right) = \log_a x\]
\[ = \frac{\log x}{\log a}\]
\[f'\left( x \right) = \frac{1}{x \log a}\]
\[\text { Since 0 < a < 1 and } x > 0, f'\left( x \right) = \frac{1}{x \log a} < 0 . \]
\[\text { So,}f\left( x \right) \text { is decreasing for all } x > 0 .\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
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) = x3 − 12x2 + 36x + 17 ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
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).
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
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
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
The function f(x) = x9 + 3x7 + 64 is increasing on
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
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 ______.
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
Which of the following functions is decreasing on `(0, pi/2)`?
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The function f (x) = 2 – 3 x is ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
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).
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
