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
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in 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) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function 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) ?
Show that the function f given by f(x) = 10x is increasing for all x ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
The function f(x) = cot−1 x + x increases in the interval
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
Function f(x) = loga x is increasing on R, if
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
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).
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the values of x for which Revenue is increasing.
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
Find the values of x, for which the function f(x) = x3 + 12x2 + 36ЁЭСе + 6 is monotonically decreasing
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
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 ______.
The slope of tangent at any point (a, b) is also called as ______.
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
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 ______.
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) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
A function f is said to be increasing at a point c if ______.
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 ______.
As one moves from left to right on a graph, what does a decrease in the \[y\]-values indicate?
For \[f'(x)=12(x-3)(x+2)\], what are the critical points obtained from \[f'(x)=0\]?
For \[f(x)=4x^3-6x^2-72x+30\], on which intervals is \[f\] increasing?
