Advertisements
Advertisements
Question
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
Advertisements
Solution
f(x) = `(x - 2)/(x + 1)` for x ≠ – 1
For function to be increasing, f'(x) > 0
Then, f'(x) = `((x + 1)"d"/("d"x)(x - 2) - (x - 2)"d"/("d"x)(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 intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, 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 intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
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 f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
The function f(x) = x3 - 3x 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 function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
y = log x satisfies for x > 1, the inequality ______.
Let f(x) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
A function \[f\] is monotonic in an interval when it is which of the following?
For \[f'(x)=12(x-3)(x+2)\], what are the critical points obtained from \[f'(x)=0\]?
For \[f(x)=4x^3-6x^2-72x+30\], on which intervals is \[f\] increasing?
