Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = 8 + 36x + 3 x^2 - 2 x^3 \]
\[f'\left( x \right) = 36 + 6x - 6 x^2 \]
\[ = - 6 \left( x^2 - x - 6 \right)\]
\[ = - 6 \left( x - 3 \right)\left( x + 2 \right)\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow - 6 \left( x - 3 \right)\left( x + 2 \right) > 0 \]
\[ \Rightarrow \left( x - 3 \right)\left( x + 2 \right) < 0 \left[ \text { Since } - 6 < 0, - 6 \left( x - 3 \right)\left( x + 2 \right) > 0 \Rightarrow \left( x - 3 \right)\left( x + 2 \right) < 0 \right]\]
\[ \Rightarrow - 2 < x < 3\]
\[ \Rightarrow x \in \left( - 2, 3 \right)\]
\[\text { So,}f(x)\text { is increasing on} \left( - 2, 3 \right) . \]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow - 6 \left( x - 3 \right)\left( x + 2 \right) < 0\]
\[ \Rightarrow \left( x - 3 \right)\left( x + 2 \right) > 0 \left[ \text { Since } - 6 < 0, - 6 \left( x - 3 \right)\left( x + 2 \right) < 0 \Rightarrow \left( x - 3 \right)\left( x + 2 \right) > 0 \right]\]
\[ \Rightarrow x < - 2 \ or \ x > 3 \]
\[ \Rightarrow x \in \left( - \infty , - 2 \right) \cup \left( 3, \infty \right)\]
\[\text { So },f(x)\text { is decreasing on } \left( - \infty , - 2 \right) \cup \left( 3, \infty \right) .\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
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.
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
Find the interval in which the following function are increasing or decreasing f(x) = x2 + 2x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 − 9x − x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?
Show that f(x) = e2x is increasing on R.
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
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
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
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 functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
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 f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly 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`
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
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
Show that function f(x) = tan x is increasing in `(0, π/2)`.
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 ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
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?
