Advertisements
Advertisements
प्रश्न
Function f(x) = | x | − | x − 1 | is monotonically increasing when
विकल्प
x < 0
x > 1
x < 1
0 < x < 1
Advertisements
उत्तर
0 < x < 1
\[f\left( x \right) = \left| x \right| - \left| x - 1 \right|\]
\[\text { Case 1: Let }x < 0 \]
\[\text { If x < 0 , then }\left| x \right| = - x\]
\[ \Rightarrow \left| x - 1 \right| = - \left( x - 1 \right)\]
\[\text { Now,}\]
\[f\left( x \right) = \left| x \right| - \left| x - 1 \right|\]
\[ = - x - \left( - x + 1 \right)\]
\[ = - x + x - 1\]
\[ = - 1\]
\[f'\left( x \right) = 0\]
\[\text { So,f }\left( x \right) \text { is not monotonically increasing when x< 0.}\]
\[\text { Case 2: Let }0 < x < 1\]
\[\text { Here,} \]
\[\left| x \right| = x\]
\[ \Rightarrow \left| x - 1 \right| = - \left( x - 1 \right)\]
\[\text { Now,}\]
\[f\left( x \right) = \left| x \right| - \left| x - 1 \right|\]
\[ = x + x - 1\]
\[ = 2x - 1\]
\[f'\left( x \right) = 2 > 0\]
\[\text { So },f\left( x \right) \text { is monotonically increasing when }0 < x < 1 . \]
\[\text { Case 3: Let x > 1} \]
\[\text { Ifx > 0, then }\left| x \right| = x\]
\[ \Rightarrow \left| x - 1 \right| = \left( x - 1 \right)\]
\[\text { Now,}\]
\[f\left( x \right) = \left| x \right| - \left| x - 1 \right|\]
\[ = x - x + 1\]
\[ = 1\]
\[f'\left( x \right) = 0\]
\[\text { So },f\left( x \right)\text { is not monotonically increasing when x >1 }.\]
\[\text { Thus },f\left( x \right) \text { is monotonically increasing when 0 < x < 1} . \]
APPEARS IN
संबंधित प्रश्न
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.
Prove that the function f given by f(x) = log cos x is strictly decreasing on `(0, pi/2)` and strictly increasing on `((3pi)/2, 2pi).`
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) = x3 − 6x2 − 36x + 2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
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 interval of increase of the function f(x) = x − ex + tan (2π/7) is
The function f(x) = cot−1 x + x increases in the interval
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
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) = x3 – 6x2 + 12x – 16, x ∈ R.
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.
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 ______.
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
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
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
The function `1/(1 + x^2)` is increasing in the interval ______
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
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) = `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 ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
