Advertisements
Advertisements
Question
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
Options
k < 3
k ≤ 3
k > 3
k ≥ 3
Advertisements
Solution
k > 3
\[f\left( x \right) = k x^3 - 9 x^2 + 9x + 3\]
\[f'\left( x \right) = 3k x^2 - 18x + 9\]
\[ = 3 \left( k x^2 - 6x + 3 \right)\]
\[\text { Given:f(x) is monotonically increasing in every interval }.\]
\[ \Rightarrow f'\left( x \right) > 0\]
\[ \Rightarrow 3 \left( k x^2 - 6x + 3 \right) > 0\]
\[ \Rightarrow \left( k x^2 - 6x + 3 \right) > 0\]
\[ \Rightarrow k > 0 \text { and } \left( - 6 \right)^2 - 4\left( k \right)\left( 3 \right) < 0 \left[ \because a x^2 + bx + c > 0 \Rightarrow a > 0 \text { and Disc} < 0 \right]\]
\[ \Rightarrow k > 0 \text { and } \left( - 6 \right)^2 - 4\left( k \right)\left( 3 \right) < 0\]
\[ \Rightarrow k > 0 \text { and }36 - 12k < 0\]
\[ \Rightarrow k > 0 \text { and }12k > 36\]
\[ \Rightarrow k > 0 \text { and } k > 3\]
\[ \Rightarrow k > 3\]
APPEARS IN
RELATED QUESTIONS
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
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 \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
The function f(x) = cot−1 x + x increases in the interval
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
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 the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is 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`
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
The function f(x) = sin x + 2x is ______
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
Show that function f(x) = tan x is increasing in `(0, π/2)`.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
The function f(x) = x3 + 3x is increasing in interval ______.
