Advertisements
Advertisements
प्रश्न
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Advertisements
उत्तर
\[f\left( x \right) = \sin x - ax + 4\]
\[f'\left( x \right) = \cos x - a\]
\[\text { Given }:f(x) \text { is increasing on R }.\]
\[ \Rightarrow f'\left( x \right) > 0\]
\[ \Rightarrow \cos x - a > 0\]
\[ \Rightarrow \cos x > a \]
\[\text { We know,}\]
\[\cos x \geq - 1, \forall x \in R \]
\[ \Rightarrow a < - 1\]
\[ \Rightarrow a \in \left( - \infty , - 1 \right)\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Every invertible function is
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure Ec of a person with income I is given as Ec = ( 0.0003 ) I2 + ( 0.075 ) I when I = 1000.
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 increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the values of x for which Revenue is increasing.
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
The function f(x) = sin x + 2x is ______
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
The function f(x) = tanx – x ______.
Find the value of x for which the function f(x)= 2x3 – 9x2 + 12x + 2 is decreasing.
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = `squarex^2 - square + square`
∴ f'(x) = `6(x - 1)(square)`
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < `square` and x > `square`
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > `square` and x < `square`
1 < `square` < 2
f(x) is decreasing if and only if x ∈ `square`
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 ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
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)
The function f(x) = x3 + 3x is increasing in interval ______.
As one moves from left to right on a graph, what do constant \[y\]-values indicate?
