Advertisements
Advertisements
Question
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
Advertisements
Solution
Given that: f(x) = `sqrt(3)` sinx – cosx – 2ax + b, a ≥ 1
Differentiating both sides w.r.t. x, we get
f'(x) = `sqrt(3) cos x + sin x - 2"a"`
For decreasing function, f'(x) < 0
∴ `sqrt(3) cos x + sin x - 2"a" < 0`
⇒ `2(sqrt(3)/2 cos x + 1/2 sin x) - 2"a" < 0`
⇒ `sqrt(3)/2 cos x + 1/2 sin x - "a" < 0`
⇒ `(cos pi/6 cos x + sin pi/6 sin x) - "a" < 0`
⇒ `cos(x - pi/6) - "a " < 0`
Since cos x ∈ [– 1, 1] and a ≥ 1
∴ f'(x) < 0
Hence, the given function is decreasing in R.
APPEARS IN
RELATED QUESTIONS
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
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.
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 − 9x − x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
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 ?
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) = e2x is increasing on R.
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
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`
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 .
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
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) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
For \[f(x)=x^3-3x^2+4x\], what is \[f'(x)\]?
Which statement about \[f'(x)=0\] and constant behaviour is correct?
