Advertisements
Advertisements
प्रश्न
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
Advertisements
उत्तर
\[f\left( x \right) = x + \cos x + ax + b\]
\[f'\left( x \right) = 1 - \sin x + a\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 1 - \sin x + a > 0\]
\[ \Rightarrow \sin x < 1 + a\]
\[\text { We know that the maximum value of sin x is 1 }.\]
\[ \Rightarrow 1 + a > 1\]
\[ \Rightarrow a > 0\]
\[ \Rightarrow a \in \left( 0, \infty \right)\]
APPEARS IN
संबंधित प्रश्न
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
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) = 2x3 + 9x2 + 12x + 20 ?
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = e2x is increasing on R.
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
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) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
Every invertible function is
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 the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Show that f(x) = x – cos x is increasing for all x.
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.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Show that f(x) = x – cos x is increasing for all x.
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
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 area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
After the domain has been divided into intervals, which quantity must be determined in each interval?
