Advertisements
Advertisements
प्रश्न
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Advertisements
उत्तर
We know, y = [x (x – 2)]² = x² (x + 4 – 4x)
= x4 – 4x3 + 4x2
On differentiating with respect to x,
`dy/dx` = 4x2 - 12x2 + 8x
= 4x (x2 - 3x + 2)
= 4x ( x - 1) (x - 2)
`dy/dx` = 0
`=>` 4x ( x - 1) (x - 2) = 0
`therefore` x = 0, 1, 2
∴ Four parts of real number line from x = 0, x = 1, x = 2 are intervals.
(`- infty`, 0), (0, 1), (1, 2), (2, 2) are formed.
| Interval | (∞, 0) | (0, 1) | (1, 2) | (2, ∞) |
| Sign of x | -ve | +ve | +ve | +ve |
| sign of (x - 1) | -ve | - ve | +ve | +ve |
| sign of (x - 2) | -ve | - ve | -ve | +ve |
| sign of `dy/dx` | -ve | +ve | -ve | +ve |
| nature of function | decreasing | increasing | decreasing | increasing |
∴ y is an increasing function in (0, 1) ∪ (2, ∞)
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
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.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
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) = x8 + 6x2 ?
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
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) ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
The function f(x) = cot−1 x + x increases in the interval
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
Every invertible function is
Function f(x) = loga x is increasing on R, if
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
A ladder 20 ft Jong leans against a vertical wall. The top-end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is ______
The function f(x) = x3 - 3x is ______.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
If f(x) = x + cosx – a then ______.
y = log x satisfies for x > 1, the inequality ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
