Advertisements
Advertisements
प्रश्न
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
Advertisements
उत्तर
\[f\left( x \right) = a \left( x + \sin x \right) + a\]
\[f'\left( x \right) = a \left( 1 + \cos x \right)\]
\[\text { For }f(x)\text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow a \left( 1 + \cos x \right) > 0 . . . \left( 1 \right)\]
\[\text { We know,}\]
\[ - 1 \leq \cos x \leq 1, \forall x \in R\]
\[ \Rightarrow 0 \leq \left( 1 + \cos x \right) \leq 2, \forall x \in R\]
\[\therefore a > 0 \left[ \text { From eq }. \left( 1 \right) \right]\]
\[ \Rightarrow a \in \left( 0, \infty \right)\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
Find the interval in which the following function are increasing or decreasing f(x) = x2 + 2x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 12x2 + 18x + 15 ?
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) = (x − 1) (x − 2)2 ?
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) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
Let f(x) = x3 − 6x2 + 15x + 3. Then,
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
Find `dy/dx,if e^x+e^y=e^(x-y)`
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
The function f(x) = x3 - 3x is ______.
The function f(x) = sin x + 2x is ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
For every value of x, the function f(x) = `1/7^x` is ______
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
Show that function f(x) = tan x is increasing in `(0, π/2)`.
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
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))` = ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
