Advertisements
Advertisements
Question
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Advertisements
Solution
\[f\left( x \right) = \log_a x\]
\[\text { Let } x_1 , x_2 \in \left( 0, \infty \right) \text { such that } x_1 < x_2 . \]
\[\text { Since given function is logarithmic, either a }> 1 or 0 < a < 1 . \]
\[\text { Case 1: Let }a > 1\]
\[\text { Here },\]
\[ x_1 < x_2 \]
\[ \Rightarrow \log_a x_1 < \log_a x_2 \]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So,f}\left( x \right)\text { is increasing on }\left( 0, \infty \right).\]
\[\text { Case 2: Let }0 < a < 1\]
\[\text { Here },\]
\[ x_1 < x_2 \]
\[ \Rightarrow \log_a x_1 > \log_a x_2 \]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right)\]
\[ \therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) > f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { Thus, for } a > 1, f(x)\text { is increasing in its domain } . \]
APPEARS IN
RELATED QUESTIONS
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
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) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
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) ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
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 decreasing on R ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
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
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
Function f(x) = ax is increasing on R, if
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
The function f(x) = sin x + 2x is ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
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 ____________.
The function f(x) = sin4x + cos4x is an increasing function if ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
