Advertisements
Advertisements
Question
Prove that the logarithmic function is strictly increasing on (0, ∞).
Advertisements
Solution
f(x) = log x
f'(x) = `1/x > 0`
`x in (0, infty)`
Hence the function `(0, infty)` is increasing.
APPEARS IN
RELATED QUESTIONS
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
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 a decreasing function on R ?
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) = 5x3 − 15x2 − 120x + 3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 107 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
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) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
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 ?
The function f(x) = cot−1 x + x increases in the interval
Let f(x) = x3 − 6x2 + 15x + 3. Then,
Every invertible function is
Function f(x) = ax is increasing on R, if
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
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
The function f(x) = 9 - x5 - x7 is decreasing for
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
y = log x satisfies for x > 1, the inequality ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
