Advertisements
Advertisements
प्रश्न
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
Advertisements
उत्तर
\[f\left( x \right) = kx - \sin x\]
\[f'\left( x \right) = k - \cos x\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow k - \cos x > 0\]
\[ \Rightarrow \cos x < k\]
\[\text { We know that the maximum value of cos x is 1 }.\]
\[\text { Since cos x<k,the minimum value of k is 1 }.\]
\[\Rightarrow k \in \left( 1, \infty \right)\]
APPEARS IN
संबंधित प्रश्न
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Prove that the logarithmic function is strictly increasing on (0, ∞).
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
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.
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
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) = 10 − 6x − 2x2 ?
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 \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
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] ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on 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 ?
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
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
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.
Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
The function f(x) = x3 - 3x is ______.
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
In case of decreasing functions, slope of tangent and hence derivative is ____________.
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`
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
Let \[f\] be continuous on \[[a,b]\] and differentiable on \[(a,b)\]. If \[f'(x)\geq0\] for every \[x\in(a,b)\], what follows?
After the domain has been divided into intervals, which quantity must be determined in each interval?
