Advertisements
Advertisements
Question
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.
Solution
We have `f (x) = x + 1/x, x in I`
Differentiating w.r.t.x, we get
`f' (x) = 1 - 1/x^2 = (x^2 - 1)/x^2`
`x^2 > 0 (1, 1), x^2 - 1 > 0 = x^2 > 1`
= `x < - 1 or x > 1`
= `x in (-oo, -1) or x in (1, oo)`
= `x in (-oo, -1) cup (1, oo) `
= `x in R - (-1, 1)`
= f (x) is strictly increasing on I
(∵ I is an interval which is a subset of R - (-1, 1))
APPEARS IN
RELATED QUESTIONS
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Prove that the function f given by f(x) = log sin x is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi)`
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
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) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
Function f(x) = cos x − 2 λ x is monotonic decreasing when
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
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.
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
Test whether the following functions are increasing or decreasing : f(x) = `(1)/x`, x ∈ R , x ≠ 0.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, 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] ______
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, 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 interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
