Advertisements
Advertisements
प्रश्न
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Advertisements
उत्तर
\[f\left( x \right) = \log_a x\]
\[\text { Let } x_1 , x_2 \in \left( 0, \infty \right) \text { such that } x_1 < x_2 . \]
\[\text { Case 1: Let a } > 1\]
\[\text{ Here },\]
\[ x_1 < x_2 \]
\[ \Rightarrow \log_a x_1 < \log_a x_2 \]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]
\[ \therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So,}f\left( x \right) \text { is increasing on }\left( 0, \infty \right).\]
\[\text { Case 2: Let }0 < a < 1\]
\[\text { Here },\]
\[ x_1 < x_2 \]
\[ \Rightarrow \log_a x_1 > \log_a x_2 \]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right)\]
\[ \therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) > f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So },f\left( x \right)\text { is decreasing on }\left( 0, \infty \right).\]
APPEARS IN
संबंधित प्रश्न
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
Find the interval in which the following function are increasing or decreasing f(x) = x2 + 2x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)2 ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Show that the function f given by f(x) = 10x is increasing for all x ?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing 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
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. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
For every value of x, the function f(x) = `1/7^x` is ______
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The function f(x) = tanx – x ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
Which of the following graph represent the strictly increasing function.
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
