Advertisements
Advertisements
प्रश्न
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Advertisements
उत्तर
f(x) = `3x + (1)/(3x)`
∴ f'(x) = `3d/dx(x) + (1)/(3)d/dx(x^-1)`
= `3 xx 1 + (1)/(3)(-1) x^-2`
= `3 - (1)/(3x^2)`
Now, f is increasing if f'(x) > 0 and is decreasing if f'(x) < 0.
Let `x ∈ (1/3, 3)`.
Then `(1)/(3) < x < 1`
∴ `(1)/(9) < x^2 < 1`
∴ `(1)/(3) < 3x^2 < 3`
∴ `3 >(1)/(3x^2) > (1)/(3)`
∴ `-3 < - (1)/(3x^2) < - (1)/(3)`
∴ `3 - 3 < 3 - (1)/(3x^2) < 3 - (1)/(3)`
∴ `0 < f'(x) < (8)/(3)`
∴ f'(x) > 0 for all x ∈ `(1/3, 1)`
∴ f is increasing in rhe interval `(1/3, 1)`
Let x ∈ `(1/9, 1/3)`.
Then `(1)/(9) < x < (1)/(3)`
∴ `(1)/(81) < x^2 < (1)/(9)`
∴ `(1)/(27) < 3x^2 < (1)/(3)`
∴ `27 > (1)/(3x^2) > 3`
∴ `-27 < -(1)/(3x^2) < - 3`
∴ `3 - 27 < 3 - (1)/(3x^2) < 3 - 3`
∴ – 24 < f'(x) < 0
∴ f'(x) < 0 for all x ∈ `(1/9, 1/3)`
∴ f is decreasing in the interval `(1/9, 1/3)`.
APPEARS IN
संबंधित प्रश्न
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Prove that the function f given by f(x) = log cos x is strictly decreasing on `(0, pi/2)` and strictly increasing on `((3pi)/2, 2pi).`
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Find the interval in which the following function are increasing or decreasing f(x) = 5x3 − 15x2 − 120x + 3 ?
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(x) = 2x3 − 24x + 107 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Show that f(x) = e2x is increasing on R.
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
The function f(x) = x9 + 3x7 + 64 is increasing on
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.
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
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.
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
Show that f(x) = x – cos x is increasing for all x.
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
The function f(x) = sin x + 2x is ______
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Let 'a' be a real number such that the function f(x) = ax2 + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax2 – 6x + 15, x∈R has a ______.
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
A function f is said to be increasing at a point c if ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
The function f(x) = xex(1 − x), x ∈ R, is ______.
In the procedure for finding intervals, which conclusion is correct when \[f'(x)<0\]?
