Advertisements
Advertisements
प्रश्न
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
उत्तर
\[\left( iii \right) f\left( x \right) = sin x\left( 1 + cos x \right), 0 < x < \frac{\pi}{2}\]
`f( x) = sin x + sin x\cos x`
` ⇒f(x) = cos x + sin x ( - sin x \right) + cos x\cos x`
`⇒ f(x) = cos x - \sin^2 x + cos^2 x`
`⇒ f(x) = co sx + \cos^2 x - 1 + \cos^2 x`
`⇒ f(x) = 2 \cos^2 x + \cosx - 1`
`⇒ f(x) = 2 \cos^2 x + 2\cosx - \cosx - 1`
`⇒ f(x) = 2\cos x( \cos x + 1 \right) - 1\left( \cos x + 1 )`
`⇒ f(x) = ( 2\cos x - 1 cos x + 1 \right)`
For f(x) to be increasing, we must have
\[f'\left( x \right) > 0\]
`⇒ ( 2 cos x - 1 )( cos x + 1 ) > 0`
This is only possible when
`(2\cos x - 1 ) > 0 and ( cos x + 1 ) > 0`
`⇒ ( 2\cos x - 1 ) > 0 and ( cos x + 1 ) > 0`
`⇒ cos x > 1/2 and cos x > - 1`
`⇒ x ∈ ( 0, π / 3 )and x ∈ ( 0, π / 2 )`
\[So, x \in \left( 0, \frac{\pi}{3} \right)\]
∴ f(x) is increasing on (o, π / 3 )
For f(x) to be decreasing, we must have
\[f'\left( x \right) < 0\]
`⇒ ( 2 cos x - 1 )( cos x + 1 ) < 0`
This is only possible when
`( 2\cosx - 1 ) < 0 and ( cosx + 1 ) > 0`
`⇒ ( 2\cosx - 1 ) < 0 and ( cosx + 1 ) > 0`
`⇒cosx < 1/2 and cosx > - 1`
`⇒ x ∈ (π/3 ,π/2 ) and x ∈ (0 ,π/2 )`
\[So, x \in \left( \frac{\pi}{3}, \frac{\pi}{2} \right)\
` ∴ f (x) \text{ is decreasing on } (π/3 ,π/2 ) .`
APPEARS IN
संबंधित प्रश्न
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Prove that the logarithmic function is strictly increasing on (0, ∞).
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
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) = 2x3 − 24x + 107 ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
The function f(x) = xx decreases on the interval
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,
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Every invertible function is
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Function f(x) = ax is increasing on R, if
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
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 the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
The function f(x) = x3 - 3x is ______.
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
y = x(x – 3)2 decreases for the values of x given by : ______.
The function f(x) = tanx – x ______.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Function given by f(x) = sin x is strictly increasing in.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
The function f(x) = sin4x + cos4x is an increasing function if ______.
