Advertisements
Advertisements
प्रश्न
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Advertisements
उत्तर
\[\text { Here }, \]
\[f\left( x \right) = \log \sin x\]
\[\text { Domain of log sin x is}\left( 0, \pi \right).\]
\[f'\left( x \right) = \frac{1}{\sin x}\cos x\]
\[ = \cot x\]
\[\text { For x} \in \left( 0, \frac{\pi}{2} \right), \text { cot x} > 0 \left[ \because \text { Cot function is positive in first quadrant }\right]\]
\[ \Rightarrow f'\left( x \right) > 0 \]
\[\text { So,f(x)is increasing on} \left( 0, \frac{\pi}{2} \right) . \]
\[\text { For x }\in \left( \frac{\pi}{2}, \pi \right), \text { cot x }< 0 \left[ \because \text { Cot function is negative in second quadrant } \right]\]
\[ \Rightarrow f'\left( x \right) < 0 \]
\[\text { So,f(x)is decreasing on }\left( \frac{\pi}{2}, \pi \right).\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Prove that the logarithmic function is strictly increasing on (0, ∞).
The interval in which y = x2 e–x is increasing is ______.
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) = −2x3 − 9x2 − 12x + 1 ?
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(x) = x4 − 4x3 + 4x2 + 15 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Show that f(x) = tan 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 ?
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Every invertible function is
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
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 ______.
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
The function f(x) = tanx – x ______.
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 interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
y = log x satisfies for x > 1, the inequality ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The function f(x) = x3 + 3x is increasing in interval ______.
