Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\]
\[ = \frac{3 x^4 + 8 x^3 - 30 x^2 - 72x + 84}{12}\]
\[f'\left( x \right) = \frac{12 x^3 + 24 x^2 - 60x - 72}{12}\]
\[ = \left( x^3 + 2 x^2 - 5x - 6 \right)\]
\[ = \left( x + 1 \right)\left( x^2 + x - 6 \right)\]
\[ = \left( x + 1 \right)\left( x - 2 \right)\left( x + 3 \right)\]
\[\text { Here }, -1, 2 \text { and } -3 \text { are the critical points }.\]
\[\text { The possible intervals are }\left( - \infty - 3 \right),\left( - 3, - 1 \right),\left( - 1, 2 \right)and\left( 2, \infty \right).\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \left( x + 1 \right)\left( x - 2 \right)\left( x + 3 \right) > 0\]
\[ \Rightarrow x \in \left( - 3, - 1 \right) \cup \left( 2, \infty \right)\]
\[\text { So },f(x)\text { is increasing on } x \in \left( - 3, - 1 \right) \cup \left( 2, \infty \right) . \]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow \left( x + 1 \right)\left( x - 2 \right)\left( x + 3 \right) < 0\]
\[ \Rightarrow x \in \left( - \infty - 3 \right) \cup \left( - 1, 2 \right) \left[ \text { From eq }. (1) \right]\]
\[\text { So,}f(x)\text { is decreasing on x } \in \left( - \infty - 3 \right) \cup \left( - 1, 2 \right) .\]
APPEARS IN
संबंधित प्रश्न
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
The interval in which y = x2 e–x is increasing is ______.
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 \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Let f defined on [0, 1] be twice differentiable such that | f (x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Function f(x) = cos x − 2 λ x is monotonic decreasing when
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
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
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.
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
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 ______.
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`
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
The function f(x) = tanx – x ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
