Advertisements
Advertisements
प्रश्न
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
उत्तर
\[f\left( x \right) = kx - \sin x\]
\[f'\left( x \right) = k - \cos x\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow k - \cos x > 0\]
\[ \Rightarrow \cos x < k\]
\[\text { We know that the maximum value of cos x is 1 }.\]
\[\text { Since cos x<k,the minimum value of k is 1 }.\]
\[\Rightarrow k \in \left( 1, \infty \right)\]
APPEARS IN
संबंधित प्रश्न
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
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) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
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) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
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) ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
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).
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Show that f(x) = x – cos x is increasing for all x.
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 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 values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The function f(x) = tan-1 x is ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
If f(x) = x + cosx – a then ______.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
