Advertisements
Advertisements
प्रश्न
Function f(x) = cos x − 2 λ x is monotonic decreasing when
पर्याय
λ > 1/2
λ < 1/2
λ < 2
λ > 2
Advertisements
उत्तर
\[f\left( x \right) = \cos x - 2 \lambda x\]
\[f'\left( x \right) = - \sin x - 2 \lambda \]
\[\text { For f(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow - \sin x - 2 \lambda < 0\]
\[ \Rightarrow sin x + 2 \lambda > 0 \]
\[ \Rightarrow 2 \lambda > - \sin x\]
\[\text { We know that the maximum value of -sin x is 1 }.\]
\[ \Rightarrow 2 \lambda > 1\]
\[ \Rightarrow \lambda > \frac{1}{2}\]
APPEARS IN
संबंधित प्रश्न
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing 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) = 2x3 − 24x + 107 ?
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 ?
Show that f(x) = e2x is increasing on R.
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure Ec of a person with income I is given as Ec = ( 0.0003 ) I2 + ( 0.075 ) I when I = 1000.
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
Show that f(x) = x – cos x is increasing for all x.
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
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 on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
The function f(x) = x2 – 2x is increasing in the interval ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
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.
Let 'a' be a real number such that the function f(x) = ax2 + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax2 – 6x + 15, x∈R has a ______.
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))` = ______.
