Advertisements
Advertisements
Question
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Advertisements
Solution
\[f\left( x \right) = a^x \]
\[f'\left( x \right) = a^x \log a\]
\[\text { Given }:f\left( x \right) \text { is decreasing on R }.\]
\[ \Rightarrow f'\left( x \right) < 0, \forall x \in R\]
\[ \Rightarrow a^x \log a < 0, \forall x \in R\]
\[\text { Here, logaritmic function is not defined for negative values of a } . \]
\[ \Rightarrow a^x > 0 \]
\[ \therefore a^x \log a < 0 \text { can be possible when } \log a < 0, \forall x \in R . \]
\[ \Rightarrow 0 < a < 1\]
APPEARS IN
RELATED QUESTIONS
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
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) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
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.
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.
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.
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
The function f(x) = 9 - x5 - x7 is decreasing for
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
