Advertisements
Advertisements
प्रश्न
Find the value of x for which the function f(x)= 2x3 – 9x2 + 12x + 2 is decreasing.
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = `squarex^2 - square + square`
∴ f'(x) = `6(x - 1)(square)`
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < `square` and x > `square`
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > `square` and x < `square`
1 < `square` < 2
f(x) is decreasing if and only if x ∈ `square`
Advertisements
उत्तर
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = 6x2 – 18x + 12 = 6(x2 – 3x + 2)
∴ f'(x) = 6(x - 1)(x – 2)
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < 1 and x > 2
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > 1 and x < 2
∴ 1 < x < 2
∴ f(x) is decreasing if and only if x ∈ (1, 2)
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Prove that the function f given by f(x) = log cos x is strictly decreasing on `(0, pi/2)` and strictly increasing on `((3pi)/2, 2pi).`
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 \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
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
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
Find `dy/dx,if e^x+e^y=e^(x-y)`
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
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 ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
As one moves from left to right on a graph, what do constant \[y\]-values indicate?
The function \[f(x)=x^3-3x^2+4x\], \[x\in\mathbf{R}\], is
For \[f'(x)=12(x-3)(x+2)\], what is the nature of \[f\] on \[(-2,3)\]?
