Advertisements
Advertisements
प्रश्न
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Advertisements
उत्तर
\[\text { Given }: f\left( x \right) = \log_a x\]
\[\text { Domain of the given function is }\left( 0, \infty \right).\]
\[\text { Let }x_1 , x_2 \in \left( 0, \infty \right) \text { such that } x_1 < x_2 . \]
\[\text { Since the given function is logarithmic, either a } > 1 or 0 < a < 1 . \]
\[\text { Case 1: Let }a > 1\]
\[\text { Here} , \]
\[ x_1 < x_2 \]
\[ \Rightarrow \log_a x_1 < \log_a x_2 \]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So },f\left( x \right)\text { is increasing on }\left( 0, \infty \right).\]
\[\text { Case 2: Let }0 < a < 1\]
\[\text { Here, }\]
\[ x_1 < x_2 \]
\[ \Rightarrow \log_a x_1 > \log_a x_2 \]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) > f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So,}f\left( x \right) \text { is decreasing on }\left( 0, \infty \right)\]
\[\text { Thus, for }0 < a < 1,f\left( x \right)\text { is decreasing in its domain }.\]
APPEARS IN
संबंधित प्रश्न
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
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) = (x − 1) (x − 2)2 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Function f(x) = cos x − 2 λ x is monotonic decreasing when
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
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
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
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 values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
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.
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
A circular pIate is contracting at the uniform rate of 5cm/sec. The rate at which the perimeter is decreasing when the radius of the circle is 10 cm Jong is
A ladder 20 ft Jong leans against a vertical wall. The top-end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is ______
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
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)`?
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
Let `"f (x) = x – cos x, x" in "R"`, then f 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) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
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)
