Advertisements
Advertisements
प्रश्न
Function f(x) = | x | − | x − 1 | is monotonically increasing when
पर्याय
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
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 function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
The interval in which y = x2 e–x is increasing is ______.
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 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Show that f(x) = e2x is increasing on R.
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
Let f(x) = x3 − 6x2 + 15x + 3. Then,
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Function f(x) = ax is increasing on R, if
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
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).
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
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 ______.
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
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 function f(x) = 9 - x5 - x7 is decreasing for
The function f(x) = sin x + 2x is ______
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
The function f(x) = sin4x + cos4x is an increasing function if ______.
