Advertisements
Advertisements
प्रश्न
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
पर्याय
λ < 1
λ > 1
λ < 2
λ > 2
Advertisements
उत्तर
λ > 2
\[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\]
\[f'\left( x \right) = \frac{\left( \sin x + \cos x \right)\left( \lambda \cos x - 2 \sin x \right) + \left( \lambda \sin x + 2 \cos x \right)\left( \cos x - \sin x \right)}{\left( \sin x + \cos x \right)^2}\]
\[ = \frac{\lambda\cos x \sin x + \lambda \cos^2 x - 2 \sin^2 x - 2 \sin x \cos x - \lambda\sin x \cos x - 2 \cos^2 x + \lambda \sin^2 x + 2 \cos x \sin x}{\left( \sin x + \cos x \right)^2}\]
\[ = \frac{- 2 \left( \sin^2 x + \cos^2 x \right) + \lambda \left( \sin^2 x + \cos^2 x \right)}{\left( \sin x + \cos x \right)^2}\]
\[ = \frac{- 2 + \lambda}{\left( \sin x + \cos x \right)^2}\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \frac{- 2 + \lambda}{\left( \sin x + \cos x \right)^2} > 0 \]
\[ \Rightarrow \lambda - 2 > 0 \left[ \because \left( \sin x + \cos x \right)^2 > 0 \right]\]
\[ \Rightarrow \lambda > 2\]
APPEARS IN
संबंधित प्रश्न
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
Prove that the logarithmic function is strictly increasing on (0, ∞).
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
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).`
The interval in which y = x2 e–x is increasing is ______.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on 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) = 5x3 − 15x2 − 120x + 3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 107 ?
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) ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
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] ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
Let f(x) = x3 − 6x2 + 15x + 3. Then,
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
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 ______
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
The function f(x) = xex(1 − x), x ∈ R, is ______.
