Advertisements
Advertisements
Question
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Advertisements
Solution
\[f\left( x \right) = \sin \left( 2x + \frac{\pi}{4} \right)\]
\[f'\left( x \right) = 2 \cos \left( 2x + \frac{\pi}{4} \right)\]
\[\text { Here, } \]
\[\frac{3\pi}{8} < x < \frac{5\pi}{8}\]
\[ \Rightarrow \frac{3\pi}{4} < 2x < \frac{5\pi}{4}\]
\[ \Rightarrow \pi < 2x + \frac{\pi}{4} < \frac{3\pi}{2}\]
\[ \Rightarrow \ cos \left( 2x + \frac{\pi}{4} \right) < 0 \left[ \because \text { Cos function is negative inthird quadrant } \right]\]
\[ \Rightarrow 2 \cos \left( 2x + \frac{\pi}{4} \right) < 0\]
\[ \Rightarrow f'\left( x \right) < 0, \forall x \in \left( \frac{3\pi}{8}, \frac{5\pi}{8} \right)\]
\[\text { So },f\left( x \right) \text { is decreasing on }\left( \frac{3\pi}{8}, \frac{5\pi}{8} \right).\]
APPEARS IN
RELATED QUESTIONS
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
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) = 6 − 9x − x2 ?
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) = 5 + 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) = 6 + 12x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
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) ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
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 values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
Function f(x) = loga x 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)
The function f(x) = x9 + 3x7 + 64 is increasing on
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
The function f(x) = 9 - x5 - x7 is decreasing for
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
Which of the following functions is decreasing on `(0, pi/2)`?
In case of decreasing functions, slope of tangent and hence derivative is ____________.
Function given by f(x) = sin x is strictly increasing in.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
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))` = ______.
The function f(x) = x3 + 3x is increasing in interval ______.
