Advertisements
Advertisements
प्रश्न
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Advertisements
उत्तर
\[f\left( x \right) = \sin x\]
\[f'\left( x \right) = \cos x > 0 \forall x \in \left( \frac{- \pi}{2}, \frac{\pi}{2} \right) \left[ \because \text { Cos function is positive in first and fourth quadrant } \right]\]
\[\text { So,}f\left( x \right)\text { is increasing on }\left( \frac{- \pi}{2}, \frac{\pi}{2} \right).\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
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
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)2 ?
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
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).
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
Function f(x) = loga x is increasing on R, if
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
Find `dy/dx,if e^x+e^y=e^(x-y)`
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
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 decreasing function.
f(x) = x4 − 2x3 + 1
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The function f (x) = 2 – 3 x is ____________.
The function f (x) = x2, for all real x, is ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Let 'a' be a real number such that the function f(x) = ax2 + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax2 – 6x + 15, x∈R has a ______.
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 ______.
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
