Advertisements
Advertisements
प्रश्न
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
विकल्प
monotonically increasing
monotonically decreasing
not monotonic
constant
Advertisements
उत्तर
monotonically decreasing
\[ \text{If 1} < x < 2, \text { then } x > 1 \text { and }x < 2 . \]
\[ \Rightarrow x - 1 > 0 \text { and }x - 2 < 0\]
\[ \Rightarrow \left| x - 1 \right| = x - 1 \text { and }\left| x - 2 \right|=-\left( x - 2 \right)\]
\[\text { Now,}\]
\[f\left( x \right) = 2 \left| x - 1 \right| + 3 \left| x - 2 \right|\]
\[ = 2\left( x - 1 \right) - 3\left( x - 2 \right)\]
\[ = 2x - 2 - 3x + 6\]
\[ = - x + 4\]
\[f'\left( x \right) = - 1 < 0, \forall x \in \left( 1, 2 \right)\]
\[\text { So, }f\left( x \right) \text { is monotonically decreasing.}\]
APPEARS IN
संबंधित प्रश्न
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 least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
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) = 8 + 36x + 3x2 − 2x3 ?
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) = x3 − 6x2 + 9x + 15 ?
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) = 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 ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
Function f(x) = loga x is increasing on R, if
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`
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).
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
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
The slope of tangent at any point (a, b) is also called as ______.
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
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] ______
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
The function f (x) = x2, for all real x, is ____________.
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
The function f(x) = x3 + 3x is increasing in interval ______.
