Advertisements
Advertisements
प्रश्न
Prove that the function f given by f(x) = log sin x is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi)`
Advertisements
उत्तर
We have, f (x) = log (sin x)
Differentiating w.r.t x, we get
`f' (x) = 1/ (sin x) (cos x) = cot x`
As cot x >0 for all `x in (0, pi/2)` and cot x < 0
For all `x in (pi/2, pi),` Therefore, f (x) is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi).`
APPEARS IN
संबंधित प्रश्न
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
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 the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
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) = −2x3 − 9x2 − 12x + 1 ?
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) = sin x is an increasing function on (−π/2, π/2) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Function f(x) = | x | − | x − 1 | is monotonically increasing when
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
The function f(x) = x9 + 3x7 + 64 is increasing on
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
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).
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
The function `1/(1 + x^2)` is increasing in the interval ______
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
The function f(x) = sin4x + cos4x is an increasing function if ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
As one moves from left to right on a graph, what does a decrease in the \[y\]-values indicate?
After the domain has been divided into intervals, which quantity must be determined in each interval?
In the procedure for finding intervals, which conclusion is correct when \[f'(x)<0\]?
Which statement about \[f'(x)=0\] and constant behaviour is correct?
