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
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
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).`
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
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) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
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 k for which f(x) = kx − sin x is increasing on R ?
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
Function f(x) = loga x is increasing on R, if
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
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
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`
The function f(x) = 9 - x5 - x7 is decreasing for
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 ______.
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
For every value of x, the function f(x) = `1/7^x` is ______
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
The function f(x) = tan-1 x is ____________.
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.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
