Advertisements
Advertisements
Question
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Advertisements
Solution
\[f\left( x \right) = x^3 - ax\]
\[f'\left( x \right) = 3 x^2 - a\]
\[\text { Given }:f\left( x \right)\text { is increasing on R }.\]
\[ \Rightarrow f'\left( x \right) \geq 0 \forall x \in R\]
\[ \Rightarrow 3 x^2 - a \geq 0 \forall x \in R\]
\[ \Rightarrow a \leq 3 x^2 \forall x \in R\]
\[\text { The least value of } 3 x^2 \text { is } 0.\]
\[\therefore a \leq 0\]
APPEARS IN
RELATED QUESTIONS
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Function f(x) = ax is increasing on R, if
Function f(x) = loga x is increasing on R, if
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
The function f(x) = sin x + 2x is ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
For every value of x, the function f(x) = `1/7^x` is ______
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
y = x(x – 3)2 decreases for the values of x given by : ______.
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
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 ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
