Advertisements
Advertisements
प्रश्न
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
Advertisements
उत्तर
\[f\left( x \right) = \sin x + \cos x, x \in \left[ 0, \frac{\pi}{2} \right]\]
\[f'\left( x \right) = \cos x - \sin x\]
\[\text { For f(x) to be increasing, we must have}\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \cos x - \sin x > 0\]
\[ \Rightarrow \sin x < \cos x\]
\[ \Rightarrow \frac{\sin x}{\cos x} < 1\]
\[ \Rightarrow \tan x < 1\]
\[ \Rightarrow x \in [0, \frac{\pi}{4}]\]
APPEARS IN
संबंधित प्रश्न
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
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)
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
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) = 2x3 − 9x2 + 12x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)2 ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
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 ?
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
A function f is said to be increasing at a point c if ______.
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)
A function \[f\] is monotonic in an interval when it is which of the following?
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] decreasing at \[x_0\]?
Which statement about \[f'(x)=0\] and constant behaviour is correct?
