Advertisements
Advertisements
Question
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Options
increasing
decreasing
constant
none of these
Advertisements
Solution
decreasing
\[\text { Given }: f\left( x \right) = 2\left| x - 1 \right| + 3\left| x - 2 \right|\]
\[\text { If 1 < x < 2, then }\left| x - 1 \right| = x - 1 . \]
\[ \Rightarrow \left| x - 2 \right| = - \left( x - 2 \right)\]
\[\text { Now,}\]
\[f\left( x \right) = 2\left| x - 1 \right| + 3\left| x - 2 \right|\]
\[ = 2 \left( x - 1 \right) + 3 \left( - x + 2 \right)\]
\[ = 2x - 2 - 3x + 6\]
\[ = - x + 4\]
\[f'\left( x \right) = - 1 < 0\]
\[\text { So,}f\left( x \right) \text { is decreasing when 1 < x < 2 } .\]
APPEARS IN
RELATED QUESTIONS
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 function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Prove that the logarithmic function is strictly increasing on (0, ∞).
Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2 ?
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) = 2x3 − 12x2 + 18x + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π) ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
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
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) 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 ______.
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 ______.
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
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 ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
