Advertisements
Advertisements
प्रश्न
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Advertisements
उत्तर
We have:
`f(x) = 3x^4 − 4x^3 −12x^2 + 5`
`Now, f'(x) = 12x^3 − 12x^2 − 24x`
`Now, f'(x) = 0`
`⇒12x^3 −12x^2−24x = 0`
`⇒12x(x^2−x−2) = 0`
`⇒12x(x^2−2x+x−2)=0`
`⇒12x[x(x−2)+1(x−2)] = 0`
`⇒12x (x+1)(x−2)=0`
`⇒x=0 ; x = −1; x = 2`
So, the points x = −1, x = 0 and x = 2 divide the real line into four disjoint intervals, namely (−∞,−1), (−1,0), (0,2) and (2,∞).
| INTERVAL | SIGN OF f ' (x)=12x (x+1)(x −2) | NATURE OF FUNCTION |
| (−∞,−1) | (−)(−)(−)=−or<0 | Strictly decreasing |
| (−1,0) | (−)(+)(−)=+or>0 | Strictly increasing |
| (0,2) | (+)(+)(−) = − or<0 | Strictly decreasing |
| (2,∞) | (+)(+)(+) = + or >0 | Strictly increasing |
(a) The given function is strictly increasing in the intervals (−1,0) ∪ (2,∞).
(b) The given function is strictly decreasing in the intervals (−∞,−1) ∪ (0,2).
APPEARS IN
संबंधित प्रश्न
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
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) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
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 ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
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 interval of increase of the function f(x) = x − ex + tan (2π/7) is
The function f(x) = cot−1 x + x increases in the interval
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
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.
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
The total cost function for production of articles is given as C = 100 + 600x – 3x2, then the values of x for which the total cost is decreasing is ______
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 function `1/(1 + x^2)` is increasing in the interval ______
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
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 ______.
y = log x satisfies for x > 1, the inequality ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
