Advertisements
Advertisements
Question
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Solution
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
RELATED QUESTIONS
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 function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Prove that the logarithmic function is strictly increasing on (0, ∞).
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Find the intervals in which the function f given by `f(x) = (4sin x - 2x - x cos x)/(2 + cos x)` is (i) increasing (ii) decreasing.
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x2 + 2x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 12x2 + 18x + 15 ?
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) = 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) = 2x3 − 24x + 7 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
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 ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4) ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
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 ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
Let f(x) = x3 − 6x2 + 15x + 3. Then,
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
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
(a) Strictly increasing
(b) strictly decreasing
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 ______.
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 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
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The function f(x) = tan-1 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:
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is
Function given by f(x) = sin x is strictly increasing in.
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) = x + cosx – a then ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
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)
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
