Advertisements
Advertisements
प्रश्न
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) ?
Advertisements
उत्तर
\[f\left( x \right) = \log \cos x\]
\[f'\left( x \right) = \frac{1}{\cos x}\left( - \sin x \right)\]
\[ = - \tan x\]
\[\text { Now,} \]
\[x \in \left( - \frac{\pi}{2}, 0 \right)\]
\[ \Rightarrow \tan x < 0\]
\[ \Rightarrow - \tan x > 0 \]
\[ \Rightarrow f'(x) > 0\]
\[\text { So, f(x) is strictly increasing on } \left( - \frac{\pi}{2}, 0 \right) . \]
\[\text { Now,} \]
\[x \in \left( 0, \frac{\pi}{2} \right)\]
\[ \Rightarrow \tan x > 0\]
\[ \Rightarrow - \tan x < 0 \]
\[ \Rightarrow f'(x) < 0\]
\[\text { So, f(x)is strictly decreasing on }\left( 0, \frac{\pi}{2} \right).\]
APPEARS IN
संबंधित प्रश्न
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.
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 function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
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 ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
Let f defined on [0, 1] be twice differentiable such that | f (x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?
What are the values of 'a' for which f(x) = ax 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 k for which f(x) = kx − sin x 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 function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
Let f(x) = x3 − 6x2 + 15x + 3. Then,
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
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) = tan-1 x is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
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(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
