Advertisements
Advertisements
Question
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Advertisements
Solution
Consider the function f'(x) = sin3x - cos3x.
f'(x)= 3cos3x+3sin3x
=3(sin3x + cos3x)
`=3sqrt2{sin3xcos(pi/4)+cos3xsin(pi/4)}`
`=3sqrt(2){sin(3x+pi/4)}`
For the increasing interval f'(x)>0
`3sqrt2{sin(3x+pi/4)}>0`
`sin(3x+pi/4)>0`
⇒0<3x+`π/4`<π
`=>0<3x<(3pi)/4`
⇒ 0 < x < π/4
Also
`sin(3x+pi/4)>0`
when, `2pi<3x+pi/4<3pi`
=>`(7pi)/4<3x<(11pi)/4`
Therefore, intervals in which function is strictly increasing in 0 < x < π/4 and 7π/12< x <11π/12.
Similarly, for the decreasing interval f'(x)< 0.
`3sqrt2{sin(3x+pi/4)}<0`
`sin(3x+pi/4)<0`
`=>pi<3x+pi/4<2pi`
`=>(3pi)/4<3x<(7pi)/4`
⇒ π/4 < x <7π/12
Also
`sin(3x+pi/4)<0`
When
`3pi<3x+pi/4<4pi`
`=>(11pi)/4<3x<(15pi)/4`
`=>(11pi)/12
The function is strictly decreasing in `pi/4and `(11pi)/12
π4<x<7π12 and 11π12<x<π" data-mce-style="position: relative;" data-mce-tabindex="0">π4<x<7π12 and 11π12<x<π
APPEARS IN
RELATED QUESTIONS
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor 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:
10 − 6x − 2x2
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that the function f given by f(x) = log sin x is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi)`
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).
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
Find the interval in which the following function are increasing or decreasing f(x) = x2 + 2x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 12x2 + 18x + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
The function f(x) = 9 - x5 - x7 is decreasing for
A ladder 20 ft Jong leans against a vertical wall. The top-end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is ______
For every value of x, the function f(x) = `1/7^x` is ______
The function `1/(1 + x^2)` is increasing in the interval ______
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The function f (x) = 2 – 3 x is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
y = log x satisfies for x > 1, the inequality ______.
