Advertisements
Advertisements
प्रश्न
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Advertisements
उत्तर
\[f\left( x \right) = x^3 - ax\]
\[f'\left( x \right) = 3 x^2 - a\]
\[\text { Given }:f\left( x \right)\text { is increasing on R }.\]
\[ \Rightarrow f'\left( x \right) \geq 0 \forall x \in R\]
\[ \Rightarrow 3 x^2 - a \geq 0 \forall x \in R\]
\[ \Rightarrow a \leq 3 x^2 \forall x \in R\]
\[\text { The least value of } 3 x^2 \text { is } 0.\]
\[\therefore a \leq 0\]
APPEARS IN
संबंधित प्रश्न
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) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
Prove that the function f given by f(x) = log sin x is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi)`
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 ______.
Find the intervals in which the function f given by `f(x) = (4sin x - 2x - x cos x)/(2 + cos x)` is (i) increasing (ii) decreasing.
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 \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
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) ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
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.
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 intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly 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.
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
A ladder 20 ft Jong leans against a vertical wall. The top-end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is ______
The function f(x) = x3 - 3x is ______.
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
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 ______.
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
Function given by f(x) = sin x is strictly increasing in.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
