Advertisements
Advertisements
प्रश्न
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
विकल्प
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
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
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.
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
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).`
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
The interval in which y = x2 e–x is increasing is ______.
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 + 9x2 + 12x + 20 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
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 \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
The function f(x) = xx decreases on the interval
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
Find `dy/dx,if e^x+e^y=e^(x-y)`
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) 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 y = xx, (x > 0) is increasing and decreasing.
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
The function `1/(1 + x^2)` is increasing in the interval ______
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
For \[f'(x)=12(x-3)(x+2)\], what is the nature of \[f\] on \[(-2,3)\]?
