Advertisements
Advertisements
प्रश्न
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Advertisements
उत्तर
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
संबंधित प्रश्न
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly 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) = (x − 1) (x − 2)2 ?
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) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
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 ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
Let f(x) = x3 − 6x2 + 15x + 3. Then,
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
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 ______
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
The function `1/(1 + x^2)` is increasing in the interval ______
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
Let \[f\] be continuous on \[[a,b]\] and differentiable on \[(a,b)\]. If \[f'(x)\leq0\] for every \[x\in(a,b)\], what follows?
For \[f(x)=4x^3-6x^2-72x+30\], which factorization of \[f'(x)\] is correct?
