Advertisements
Advertisements
Question
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Advertisements
Solution
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = \left\{ x\left( x - 2 \right) \right\}^2 \]
\[ = \left( x^2 - 2x \right)^2 \]
\[ = x^4 + 4 x^2 - 4 x^3 \]
\[f'\left( x \right) = 4 x^3 + 8x - 12 x^2 \]
\[ = 4x \left( x^2 - 3x + 2 \right)\]
\[ = 4x \left( x - 1 \right)\left( x - 2 \right)\]
\[\text { Here, 0, 1 and 2 are the critical points}.\]
\[\text { The possible intervals are }\left( - \infty , 0 \right),\left( 0, 1 \right),\left( 1, 2 \right)\text { and }\left( 2, \infty \right).\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 4x \left( x - 1 \right)\left( x - 2 \right) > 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 2 \right) > 0\]
\[ \Rightarrow x \in \left( 0, 1 \right) \cup \left( 2, \infty \right) \]
\[\text { So,f(x)is increasing on x } \in \left( 0, 1 \right) \cup \left( 2, \infty \right) . \]
\[\text { For } f(x)\text { to be decreasing, we must have } \]
\[f'(x) < 0\]
\[ \Rightarrow 4x\left( x - 1 \right)\left( x - 2 \right) < 0\]
\[ \Rightarrow x\left( x - 1 \right)\left( x - 2 \right) < 0\]
\[ \Rightarrow x \in \left( - \infty , 0 \right) \cup \left( 1, 2 \right)\]
\[\text { So, f(x) is decreasing on x } \in \left( - \infty , 0 \right) \cup \left( 1, 2 \right) .\]
APPEARS IN
RELATED QUESTIONS
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:
−2x3 − 9x2 − 12x + 1
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
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) = 5 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 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) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
Find `dy/dx,if e^x+e^y=e^(x-y)`
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
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.
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
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.
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
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] ______
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
In case of decreasing functions, slope of tangent and hence derivative is ____________.
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is
Which of the following graph represent the strictly increasing function.
Let f(x) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
