Advertisements
Advertisements
Question
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Advertisements
Solution
f(x) =`("x - 2")/("x + 1")`, x ≠ 0
For function to be increasing, f '(x) > 0
Then f '(x) = `(("x + 1") "d"/"dx" ("x - 2") - ("x - 2") "d"/"dx" ("x" + 1))/("x + 1")^2`
`= (("x + 1") - ("x - 2"))/("x + 1")^2 = ("x" + 1 - "x" + 2)/("x" + 1)^2`
`= 3/("x + 1")^2` > 0 ....[∵ (x + 1) ≠ 0, (x + 1)2 > 0]
Thus, f(x) is an increasing function for x ≠ - 1.
APPEARS IN
RELATED QUESTIONS
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) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
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) = cos x + a2 x + b is strictly increasing on R ?
The function f(x) = cot−1 x + x increases in the interval
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure Ec of a person with income I is given as Ec = ( 0.0003 ) I2 + ( 0.075 ) I when I = 1000.
Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Show that f(x) = x – cos x is increasing for all x.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
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] ______
The function `1/(1 + x^2)` is increasing in the interval ______
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
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)
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
