Advertisements
Advertisements
प्रश्न
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Advertisements
उत्तर
y = `(4sinθ)/(2 + cosθ) - θ`
∴ `dy/"dθ" = d/"dθ"[(4sinθ)/(2 + cosθ) - θ]`
= `d/"dθ"((4sinθ)/(2 + cosθ)) - d/"dθ"(θ)`
= `((2 + cosθ).d/"dθ"(4sinθ) - 4sinθ.d/"dθ"(2 + cos θ))/((2 + cosθ)^2) - 1`
= `((2 + cosθ)"(4cosθ) - (4sinθ)(0 - sinθ))/((2 + cosθ)^2) - 1`
= `(8cosθ + 4cos^2θ + 4sin^2θ)/(2 + cosθ)^2 - 1`
= `(8cosθ + 4(cos^2θ + sin^2θ))/(2 + cosθ)^2 - 1`
= `(8cosθ + 4)/(2 + cosθ)^2 - 1`
= `((8cos θ + 4) - (2 + cosθ)^2)/(2 + cosθ)^2`
= `(8cosθ + 4 - 4 - 4cosθ - cos^2θ)/(2 + cosθ)^2`
= `(4cosθ - cos^2θ)/(2 + cosθ)^2`
= `(cosθ(4 - cosθ))/(2 + cosθ)^2`
Since, `θ ∈ [0, pi/2], cos θ ≥ 0` Also, cos θ < 4
∴ 4 - cos θ > 0
∴ cos θ (4 - cosθ) ≥ 0
∴ `(cosθ(4 - cosθ))/(2 + cosθ^2)≥ 0`
∴ `dy/"dθ" ≥ 0 "for all" θ ∈[0, pi/2]`
Hence, y is an increasing function if `θ ∈[0, pi/2]`.
APPEARS IN
संबंधित प्रश्न
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.
The interval in which y = x2 e–x is increasing is ______.
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 5x3 − 15x2 − 120x + 3 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = x3 − 15x2 + 75x − 50 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)?
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Find `dy/dx,if e^x+e^y=e^(x-y)`
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure Ec of a person with income I is given as Ec = ( 0.0003 ) I2 + ( 0.075 ) I when I = 1000.
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
The slope of tangent at any point (a, b) is also called as ______.
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
A circular pIate is contracting at the uniform rate of 5cm/sec. The rate at which the perimeter is decreasing when the radius of the circle is 10 cm Jong is
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
The function f (x) = x2, for all real x, is ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f 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 ____________.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
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`
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 ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
