Advertisements
Advertisements
प्रश्न
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
पर्याय
k < 3
k ≤ 3
k > 3
k ≥ 3
Advertisements
उत्तर
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
संबंधित प्रश्न
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
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:
−2x3 − 9x2 − 12x + 1
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 5x3 − 15x2 − 120x + 3 ?
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) = sin x is an increasing function on (−π/2, π/2) ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R ?
Function f(x) = | x | − | x − 1 | is monotonically increasing when
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).
Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Show that f(x) = x – cos x is increasing for all x.
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.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
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`
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
The function `1/(1 + x^2)` is increasing in the interval ______
Which of the following functions is decreasing on `(0, pi/2)`?
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
Which of the following graph represent the strictly increasing function.
Function given by f(x) = sin x is strictly increasing in.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
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 ______.
A function f is said to be increasing at a point c if ______.
