Advertisements
Advertisements
प्रश्न
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
Advertisements
उत्तर
(A) Let (x) = cos x, then
f' (x) = - sin x < 0 for all `x in (0, pi/2)` ....`[∴ sin x > 0 AA x in (0, pi/2)]`
(B) Let f(x) = cos 2x, then
f'(x) = -2 sin 2x < 0 for all `x in (0, pi/2)` ...(`∵ sin x > 0 (0, pi) = sin 2x > 0 (0, pi/2)`
= f is strictly decreasing on `(0, pi/2)`
(c) Let f(x) = cos 3x, then f'(x) = -3 sin 3x,
which assume +ve as well as -ve values in`(0, pi/2)`
`[0 < x < pi/2 = 0 <3x < (3pi)/2]`and `sin 3x > 0 (0, pi/2), sin 3x < o (pi, (3pi)/2)`
∴ f is neither increasing nor decreasing on `(0, pi/2)`
(D) Let f(x) = tan x, then f'(x) = sec2 x > 0 for all `x in (0, pi/2)` ....`[∵ sec^2 x > 0 AA x in (0, pi/2)]`
= f is strictly increasing on `(0, pi/2)`
Thus, we find that the function in (A) and (B) are strictly decreasing on `(0, pi/2).`
APPEARS IN
संबंधित प्रश्न
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Prove that the logarithmic function is strictly increasing on (0, ∞).
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 107 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
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) = e2x is increasing on R.
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
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).
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
The function f(x) = xx decreases on the interval
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 .
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
The slope of tangent at any point (a, b) is also called as ______.
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
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
Find the value of x for which the function f(x)= 2x3 – 9x2 + 12x + 2 is decreasing.
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = `squarex^2 - square + square`
∴ f'(x) = `6(x - 1)(square)`
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < `square` and x > `square`
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > `square` and x < `square`
1 < `square` < 2
f(x) is decreasing if and only if x ∈ `square`
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
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 ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
