Advertisements
Advertisements
प्रश्न
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
उत्तर
\[f\left( x \right) = \sin x - \cos x, x \in \left( 0, 2\pi \right)\]
\[f'\left( x \right) = \cos x + \sin x\]
\[\text { For f(x) to be increasin, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \cos x + \sin x > 0\]
\[ \Rightarrow \sin x > - \cos x\]
\[ \Rightarrow \tan x > - 1\]
\[ \Rightarrow x \in \left( 0, \frac{3\pi}{4} \right) \cup \left( \frac{7\pi}{4}, 2\pi \right)\]
\[\text { So,f(x)is increasing on } \left( 0, \frac{3\pi}{4} \right) \cup \left( \frac{7\pi}{4}, 2\pi \right) . \]
\[\text { For f(x) to be decreasing we must have},\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow \cos x + \sin x < 0\]
\[ \Rightarrow \sin x < - \cos x\]
\[ \Rightarrow \tan x < - 1\]
\[ \Rightarrow x \in \left( \frac{3\pi}{4}, \frac{7\pi}{4} \right)\]
\[\text { So,f(x)is decreasing on }\left( \frac{3\pi}{4}, \frac{7\pi}{4} \right).\]
APPEARS IN
संबंधित प्रश्न
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 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) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
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) ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
Let f(x) = x3 − 6x2 + 15x + 3. Then,
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Function f(x) = loga x is increasing on R, if
Test whether the following functions are increasing or decreasing : f(x) = `(1)/x`, x ∈ R , x ≠ 0.
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
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 following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
Which of the following functions is decreasing on `(0, pi/2)`?
The function f (x) = x2, for all real x, is ____________.
The function f(x) = tan-1 x is ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
If f(x) = x + cosx – a then ______.
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)
