Advertisements
Advertisements
प्रश्न
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Advertisements
उत्तर
\[f\left( x \right) = \cos^2 x\]
\[f'\left( x \right) = 2 \cos x \left( - \sin x \right)\]
\[ \Rightarrow f'\left( x \right) = - \sin \left( 2x \right) . . . \left( 1 \right)\]
\[\text { Now,}\]
\[0 < x < \frac{\pi}{2}\]
\[ \Rightarrow 0 < 2x < \pi \]
\[ \Rightarrow \sin 2x > 0 \left[ \because \text { Sine fuction is positive in first and second quadrant } \right]\]
\[ \Rightarrow - \sin 2x < 0\]
\[ \Rightarrow f'\left( x \right) < 0 \left[ \text { From eq.} (1) \right]\]
\[\text { So,f(x)is decreasing on}\left( 0, \frac{\pi}{2} \right).\]
APPEARS IN
संबंधित प्रश्न
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
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) = x2 + 2x − 5 ?
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) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
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
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).
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 following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
The slope of tangent at any point (a, b) is also called as ______.
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
A ladder 20 ft Jong leans against a vertical wall. The top-end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall 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 ______
Which of the following functions is decreasing on `(0, pi/2)`?
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The function f (x) = 2 – 3 x is ____________.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
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)
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
The function f(x) = xex(1 − x), x ∈ R, is ______.
For \[f'(x)=12(x-3)(x+2)\], what are the critical points obtained from \[f'(x)=0\]?
