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
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
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
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
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)2 ?
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) = sin x is an increasing function on (−π/2, π/2) ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
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 ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
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) = x3 − 9kx2 + 27x + 30 is increasing on R, then
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 total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Show that f(x) = x – cos x is increasing for all x.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
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
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 ______.
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
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 for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
Which of the following functions is decreasing on `(0, pi/2)`?
2x3 - 6x + 5 is an increasing function, if ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
A function f is said to be increasing at a point c if ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
