Advertisements
Advertisements
Question
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
Solution
\[f\left( x \right) = x - \left[ x \right]\]
\[\text { Let } x_1 , x_2 \in \left( 0, 1 \right) \text { such that } x_1 < x_2 . \text { Then }, \]
\[\left[ x_1 \right]=\left[ x_2 \right]= 0 ...(1)\]
\[\text { Now,}\]
\[ x_1 < x_2 \]
\[ \Rightarrow x_1 - \left[ x_1 \right] < x_2 - \left[ x_2 \right] \left[ \text { From eq }. (1) \right]\]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, 1 \right)\]
\[\text { So},f\left( x \right) \text { is increasing on }\left( 0, 1 \right).\]
APPEARS IN
RELATED QUESTIONS
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
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 \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4) ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
Test whether the following functions are increasing or decreasing : f(x) = `(1)/x`, x ∈ R , x ≠ 0.
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R 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 function f(x) = sin x + 2x is ______
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
