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
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Prove that the logarithmic function is strictly increasing on (0, ∞).
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
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) = x2 + 2x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 5x3 − 15x2 − 120x + 3 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?
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(x) = 2x3 − 9x2 + 12x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
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 f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Let f defined on [0, 1] be twice differentiable such that | f (x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] 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 ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
The function f(x) = cot−1 x + x increases in the interval
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
Function f(x) = ax is increasing on R, if
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
The function f(x) = x3 - 3x 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 values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
In case of decreasing functions, slope of tangent and hence derivative is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
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.
Function given by f(x) = sin x is strictly increasing in.
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 ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
