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
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)
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
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) = x3 − 6x2 − 36x + 2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?
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) = x4 − 4x ?
Find the interval in which the following function are increasing or decreasing f(x) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
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 the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
The function f(x) = xx decreases on the interval
Function f(x) = ax is increasing on R, if
Function f(x) = loga x is increasing on R, if
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
A circular pIate is contracting at the uniform rate of 5cm/sec. The rate at which the perimeter is decreasing when the radius of the circle is 10 cm Jong is
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
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
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 ______.
If f(x) = x + cosx – a then ______.
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 intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
