Advertisements
Advertisements
Question
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Advertisements
Solution
`f(x) = x - 1/x`
`f'(x) = d/dx (x-1/x)`
`= 1-(-1/x^2)`
`= 1+1/x^2 > 0` for all x ∈ R, where x ≠ 0
APPEARS IN
RELATED QUESTIONS
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
Prove that the logarithmic function is strictly increasing on (0, ∞).
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 + 9x2 + 12x + 20 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
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
Let f(x) = x3 − 6x2 + 15x + 3. Then,
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
The total cost function for production of articles is given as C = 100 + 600x – 3x2, then the values of x for which the total cost is decreasing is ______
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
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) = x3 - 3x is ______.
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
The function f(x) = tan-1 x is ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
The function f(x) = x3 + 3x is increasing in interval ______.
