Advertisements
Advertisements
प्रश्न
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Advertisements
उत्तर
\[f\left( x \right) = \cos^2 x\]
\[f'\left( x \right) = 2 \cos x \left( - \sin x \right)\]
\[ \Rightarrow f'\left( x \right) = - \sin \left( 2x \right) . . . \left( 1 \right)\]
\[\text { Now,}\]
\[0 < x < \frac{\pi}{2}\]
\[ \Rightarrow 0 < 2x < \pi \]
\[ \Rightarrow \sin 2x > 0 \left[ \because \text { Sine fuction is positive in first and second quadrant } \right]\]
\[ \Rightarrow - \sin 2x < 0\]
\[ \Rightarrow f'\left( x \right) < 0 \left[ \text { From eq.} (1) \right]\]
\[\text { So,f(x)is decreasing on}\left( 0, \frac{\pi}{2} \right).\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
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)`
Prove that the function f given by f(x) = log cos x is strictly decreasing on `(0, pi/2)` and strictly increasing on `((3pi)/2, 2pi).`
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing f(x) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
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 set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
The function f(x) = xx decreases on the interval
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
Find `dy/dx,if e^x+e^y=e^(x-y)`
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
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.
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
The function f(x) = x3 - 3x is ______.
Which of the following functions is decreasing on `(0, pi/2)`?
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
Let f(x) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
The function f(x) = sin4x + cos4x is an increasing function if ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
