Advertisements
Advertisements
प्रश्न
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Advertisements
उत्तर
\[f\left( x \right) = \sin \left( 2x + \frac{\pi}{4} \right)\]
\[f'\left( x \right) = 2 \cos \left( 2x + \frac{\pi}{4} \right)\]
\[\text { Here, } \]
\[\frac{3\pi}{8} < x < \frac{5\pi}{8}\]
\[ \Rightarrow \frac{3\pi}{4} < 2x < \frac{5\pi}{4}\]
\[ \Rightarrow \pi < 2x + \frac{\pi}{4} < \frac{3\pi}{2}\]
\[ \Rightarrow \ cos \left( 2x + \frac{\pi}{4} \right) < 0 \left[ \because \text { Cos function is negative inthird quadrant } \right]\]
\[ \Rightarrow 2 \cos \left( 2x + \frac{\pi}{4} \right) < 0\]
\[ \Rightarrow f'\left( x \right) < 0, \forall x \in \left( \frac{3\pi}{8}, \frac{5\pi}{8} \right)\]
\[\text { So },f\left( x \right) \text { is decreasing on }\left( \frac{3\pi}{8}, \frac{5\pi}{8} \right).\]
APPEARS IN
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 ?
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing 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) = x3 − 6x2 − 36x + 2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
The function f(x) = cot−1 x + x increases in the interval
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
Every invertible function is
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The function f(x) = tanx – x ______.
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
The function f (x) = 2 – 3 x is ____________.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
