Advertisements
Advertisements
प्रश्न
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
उत्तर
\[f\left( x \right) = x\left| x \right|, x \in R\]
\[\text { Case I: When x } \geq 0\]
\[f\left( x \right) = x\left| x \right| = x\left( x \right) = x^2 \]
\[ \Rightarrow f'\left( x \right) = 2x \geq 0 \forall x \geq 0\]
\[\text { So,} f\left( x \right)\text { is increasing for x } \geq 0 . \]
\[\text { Case II: When } x < 0\]
\[f\left( x \right) = x\left| x \right| = x\left( - x \right) = - x^2 \]
\[ \Rightarrow f'\left( x \right) = - 2x \geq 0 \forall x < 0\]
\[\text { So, }f\left( x \right)\text { is increasing for } x < 0 . \]
\[\text { Hence }, f\left( x \right)\text { is increasing for x } \in R . \]
APPEARS IN
संबंधित प्रश्न
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
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).`
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
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) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
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
Function f(x) = ax is increasing on R, 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 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 function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is 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 ______.
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
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
The function f(x) = sin x + 2x is ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
The function f (x) = 2 – 3 x is ____________.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Find the value of x for which the function f(x)= 2x3 – 9x2 + 12x + 2 is decreasing.
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = `squarex^2 - square + square`
∴ f'(x) = `6(x - 1)(square)`
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < `square` and x > `square`
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > `square` and x < `square`
1 < `square` < 2
f(x) is decreasing if and only if x ∈ `square`
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
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 ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
