Advertisements
Advertisements
प्रश्न
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
पर्याय
`[–1, oo)`
[– 2, – 1]
`(-oo, -2]`
[– 1, 1]
Advertisements
उत्तर
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is [– 2, – 1].
Explanation:
The given function is f(x) = 2x3 + 9x2 + 12x – 1
f'(x) = 6x2 + 18x + 12
For increasing and decreasing f'(x) = 0
∴ 6x2 + 18x + 12 = 0
⇒ x2 + 3x + 2 = 0
⇒ x2 + 2x + x + 2 = 0
⇒ x(x + 2) + 1(x + 2) = 0
⇒ (x + 2)(x + 1) = 0
⇒ x = – 2, x = – 1
The possible intervals are `(–oo, – 2), (– 2, – 1), (– 1, oo)`
Now f'(x) = (x + 2) (x + 1)
⇒ `"f'"(x)_((-oo"," -2))` = (–) (–) = (+) increasing
⇒ `"f'"(x)_((-2"," -1))` = (+) (–) = (–) decreasing
⇒ `"f'"(x)_((-1"," oo))` = (+) (+) = (+) increasing.
APPEARS IN
संबंधित प्रश्न
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
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) = 6 + 12x + 3x2 − 2x3 ?
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 \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
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) = 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 ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing 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 \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
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
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Show that f(x) = x – cos x is increasing for all x.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
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) = 2x3 – 6x2 + 6x + 24 is strictly increasing
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The function f(x) = tan-1 x is ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
y = log x satisfies for x > 1, the inequality ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
