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
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
The interval in which y = x2 e–x is increasing is ______.
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
The function f(x) = x2 e−x is monotonic increasing when
Find `dy/dx,if e^x+e^y=e^(x-y)`
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
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 increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
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
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
