Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = \left\{ x\left( x - 2 \right) \right\}^2 \]
\[ = \left( x^2 - 2x \right)^2 \]
\[ = x^4 + 4 x^2 - 4 x^3 \]
\[f'\left( x \right) = 4 x^3 + 8x - 12 x^2 \]
\[ = 4x \left( x^2 - 3x + 2 \right)\]
\[ = 4x \left( x - 1 \right)\left( x - 2 \right)\]
\[\text { Here, 0, 1 and 2 are the critical points}.\]
\[\text { The possible intervals are }\left( - \infty , 0 \right),\left( 0, 1 \right),\left( 1, 2 \right)\text { and }\left( 2, \infty \right).\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 4x \left( x - 1 \right)\left( x - 2 \right) > 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 2 \right) > 0\]
\[ \Rightarrow x \in \left( 0, 1 \right) \cup \left( 2, \infty \right) \]
\[\text { So,f(x)is increasing on x } \in \left( 0, 1 \right) \cup \left( 2, \infty \right) . \]
\[\text { For } f(x)\text { to be decreasing, we must have } \]
\[f'(x) < 0\]
\[ \Rightarrow 4x\left( x - 1 \right)\left( x - 2 \right) < 0\]
\[ \Rightarrow x\left( x - 1 \right)\left( x - 2 \right) < 0\]
\[ \Rightarrow x \in \left( - \infty , 0 \right) \cup \left( 1, 2 \right)\]
\[\text { So, f(x) is decreasing on x } \in \left( - \infty , 0 \right) \cup \left( 1, 2 \right) .\]
APPEARS IN
संबंधित प्रश्न
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
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(x) = x4 − 4x ?
Show that f(x) = e2x is increasing on R.
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
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) ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
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
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Find `dy/dx,if e^x+e^y=e^(x-y)`
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 values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
For every value of x, the function f(x) = `1/7^x` is ______
Function given by f(x) = sin x is strictly increasing in.
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
