Advertisements
Advertisements
प्रश्न
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Advertisements
उत्तर
f(x) = - 2x3 - 9x2 - 12x + 1
f'(x) = -6x2 - 18x - 12 = - 6(x2 + 3x + 2)
= - 6(x + 2)(x + 1)
If f'(x) = 0
-6(x + 2)(x + 1) = 0
x = - 2, -1 divides the real line into three intervals: `(- infty, -2), (-2, -1)` and `(-1, infty)`.
The function f is continuously increasing in the intervals `(- infty, -2)` and `(-1, infty)` and continuously decreasing in (-2, -1).
APPEARS IN
संबंधित प्रश्न
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
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 ?
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
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 \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
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).
What are the values of 'a' for which f(x) = ax is increasing on R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
The function f(x) = xx decreases on the interval
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
The function f(x) = x9 + 3x7 + 64 is increasing on
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the values of x for which Revenue is increasing.
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
The function f(x) = 9 - x5 - x7 is decreasing for
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
The function `1/(1 + x^2)` is increasing in the interval ______
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
Which of the following functions is decreasing on `(0, pi/2)`?
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
Find the value of x for which the function f(x)= 2x3 – 9x2 + 12x + 2 is decreasing.
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = `squarex^2 - square + square`
∴ f'(x) = `6(x - 1)(square)`
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < `square` and x > `square`
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > `square` and x < `square`
1 < `square` < 2
f(x) is decreasing if and only if x ∈ `square`
If f(x) = x + cosx – a then ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
