Advertisements
Advertisements
Question
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Advertisements
Solution
\[f\left( x \right) = \cos^2 x\]
\[f'\left( x \right) = 2 \cos x \left( - \sin x \right)\]
\[ \Rightarrow f'\left( x \right) = - \sin \left( 2x \right) . . . \left( 1 \right)\]
\[\text { Now,}\]
\[0 < x < \frac{\pi}{2}\]
\[ \Rightarrow 0 < 2x < \pi \]
\[ \Rightarrow \sin 2x > 0 \left[ \because \text { Sine fuction is positive in first and second quadrant } \right]\]
\[ \Rightarrow - \sin 2x < 0\]
\[ \Rightarrow f'\left( x \right) < 0 \left[ \text { From eq.} (1) \right]\]
\[\text { So,f(x)is decreasing on}\left( 0, \frac{\pi}{2} \right).\]
APPEARS IN
RELATED QUESTIONS
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Prove that the logarithmic function is strictly increasing on (0, ∞).
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Prove that the function f(x) = loge x is increasing on (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) = −2x3 − 9x2 − 12x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
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).
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
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) = xx decreases on the interval
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
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 values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The function f(x) = tanx – x ______.
The function f (x) = x2, for all real x, is ____________.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
Which of the following graph represent the strictly increasing function.
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
