Advertisements
Advertisements
प्रश्न
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).
Advertisements
उत्तर
\[f\left( x \right) = \cos x\]
\[f'\left( x \right) = - \sin x\]
\[\left( i \right) \] \[\text { Here },\]
\[0 < x < \pi\]
\[ \Rightarrow \sin x > 0 \left[ \because \text { Sine function is positive in first and second quadrant } \right]\]
\[ \Rightarrow - \sin x < 0\]
\[ \Rightarrow f'\left( x \right) < 0, \forall x \in \left( 0, \pi \right)\]
\[\text { So, f(x) is strictly decreasing on } \left( 0, \pi \right) . \]
\[\left( ii \right) \] \[\text { Here, }\]
\[\pi < x < 2\pi\]
\[ \Rightarrow \sin x < 0 \left[ \because \text { Sine function is negative in third and fourth quadrant} \right]\]
\[ \Rightarrow - \sin x > 0\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in \left( \pi, 2\pi \right)\]
\[\text { So,f(x)is strictly increasing on } \left( \pi, 2\pi \right) . \]
\[\left( iii \right) \] \[\text { From eqs. (1) and (2), we get }\]
\[f(x)\text { is strictly decreasing on } \left( 0, \pi \right) \text { and is strictly increasing on } \left( \pi, 2\pi \right) . \]
\[\text { So,}f\left( x \right) \text { is neither increasing nor decreasing on}\left( 0, 2\pi \right).\]
APPEARS IN
संबंधित प्रश्न
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
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 \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
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) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
The function f(x) = x9 + 3x7 + 64 is increasing on
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. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
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.
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).
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 increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
Show that f(x) = x – cos x is increasing for all x.
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
A circular pIate is contracting at the uniform rate of 5cm/sec. The rate at which the perimeter is decreasing when the radius of the circle is 10 cm Jong is
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 ______
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
The function f(x) = tan-1 x is ____________.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
