Advertisements
Advertisements
प्रश्न
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
विकल्प
a = b
\[a = \frac{1}{2}b\]
\[a \leq - \frac{1}{2}\]
\[a > - \frac{3}{2}\]
Advertisements
उत्तर
\[a \leq - \frac{1}{2}\]
\[Given: f\left( x \right) = \cos \left| x \right| - 2ax + b\]
\[\text { Now}, \left| x \right| =\begin{cases} x ,& x \geq 0 \\ - x, & x < 0 \end{cases}\]
\[\text { and } \cos \left| x \right| = \begin{cases} \cos\left( x \right) , & x \geq 0 \\cos\left( - x \right) = cos\left( x \right), & x < 0\end{cases}\]
\[ \therefore \cos\left| x \right| = \cos x , \forall x \in R\]
\[ \therefore f\left( x \right) = \cos x - 2ax + b\]
\[ \Rightarrow f'\left( x \right) = - \sin x - 2a\]
\[\text { It is given that f(x) is increasing } . \]
\[ \Rightarrow f'\left( x \right) \geq 0\]
\[ \Rightarrow - \sin x - 2a \geq 0\]
\[ \Rightarrow \sin x + 2a \leq 0\]
\[ \Rightarrow 2a \leq - \sin x\]
\[\text { The least value of -sin x is -1 }.\]
\[ \Rightarrow 2a \leq - 1\]
\[ \Rightarrow a \leq \frac{- 1}{2}\]
APPEARS IN
संबंधित प्रश्न
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that the logarithmic function is strictly increasing on (0, ∞).
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
Find the interval in which the following function are increasing or decreasing f(x) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Function f(x) = ax is increasing on R, if
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
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).
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) = x3 – 6x2 + 12x – 16, x ∈ R.
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
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 increasing function.
f(x) = x2 + 2x - 5
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the values of x for which Revenue is increasing.
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
The function `1/(1 + x^2)` is increasing in the interval ______
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The function f(x) = sin4x + cos4x is an increasing function if ______.
The function f(x) = xex(1 − x), x ∈ R, is ______.
