Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\]
\[f'\left( x \right) = 12 x^3 - 12 x^2 - 24x\]
\[ = 12x\left( x^2 - x - 2 \right)\]
\[ = 12x\left( x + 1 \right)\left( x - 2 \right)\]
\[\text { Here }, x = 0,x = - 1\text { and x = 2 are the critical points }.\]
\[\text { The possible intervals are }\left( - \infty , - 1 \right),\left( - 1, 0 \right),\left( 0, 2 \right)\text { and }\left( 2, \infty \right). .....(1)\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 12x\left( x + 1 \right)\left( x - 2 \right) > 0 \left[ \text { Since }, 12 > 0, 12x\left( x + 1 \right)\left( x - 2 \right) > 0 \Rightarrow x\left( x + 1 \right)\left( x - 2 \right) > 0 \right]\]
\[ \Rightarrow x\left( x + 1 \right)\left( x - 2 \right) > 0\]
\[ \Rightarrow x \in \left( - 1, 0 \right) \cup \left( 2, \infty \right) \left[ \text { From eq. } (1) \right]\]
\[\text { So,f(x)is increasing on x } \in \left( - 1, 0 \right) \cup \left( 2, \infty \right) .\]
\[\text { For f(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 12x\left( x + 1 \right)\left( x - 2 \right) < 0 \left[ \text { Since }12 > 0, 12x\left( x + 1 \right)\left( x - 2 \right) < 0 \Rightarrow x\left( x + 1 \right)\left( x - 2 \right) < 0 \right]\]
\[ \Rightarrow x \left( x + 1 \right)\left( x - 2 \right) < 0\]
\[ \Rightarrow x \in \left( - \infty , - 1 \right) \cup \left( 0, 2 \right) \left[ \text { From eq. } (1) \right]\]
\[\text { So,f(x)is decreasing on x }\in \left( - \infty , - 1 \right) \cup \left( 0, 2 \right) .\]
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 following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Prove that the logarithmic function is strictly increasing on (0, ∞).
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
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] ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Function f(x) = | x | − | x − 1 | is monotonically increasing when
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.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
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
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
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 ______
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
y = x(x – 3)2 decreases for the values of x given by : ______.
The function f(x) = tan-1 x is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Which of the following graph represent the strictly increasing function.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
The function f(x) = sin4x + cos4x is an increasing function if ______.
