Advertisements
Advertisements
Question
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).
Advertisements
Solution
\[f\left( x \right) = \cos x\]
\[f'\left( x \right) = - \sin x\]
\[\left( i \right) \] \[\text { Here },\]
\[0 < x < \pi\]
\[ \Rightarrow \sin x > 0 \left[ \because \text { Sine function is positive in first and second quadrant } \right]\]
\[ \Rightarrow - \sin x < 0\]
\[ \Rightarrow f'\left( x \right) < 0, \forall x \in \left( 0, \pi \right)\]
\[\text { So, f(x) is strictly decreasing on } \left( 0, \pi \right) . \]
\[\left( ii \right) \] \[\text { Here, }\]
\[\pi < x < 2\pi\]
\[ \Rightarrow \sin x < 0 \left[ \because \text { Sine function is negative in third and fourth quadrant} \right]\]
\[ \Rightarrow - \sin x > 0\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in \left( \pi, 2\pi \right)\]
\[\text { So,f(x)is strictly increasing on } \left( \pi, 2\pi \right) . \]
\[\left( iii \right) \] \[\text { From eqs. (1) and (2), we get }\]
\[f(x)\text { is strictly decreasing on } \left( 0, \pi \right) \text { and is strictly increasing on } \left( \pi, 2\pi \right) . \]
\[\text { So,}f\left( x \right) \text { is neither increasing nor decreasing on}\left( 0, 2\pi \right).\]
APPEARS IN
RELATED QUESTIONS
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
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:
(x + 1)3 (x − 3)3
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
Find the interval in which the following function are increasing or decreasing f(x) = x2 + 2x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
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) = x − [x] is increasing in (0, 1) ?
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
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) = | x | − | x − 1 | is monotonically increasing when
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
Find `dy/dx,if e^x+e^y=e^(x-y)`
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
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 the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
The function f(x) = 9 - x5 - x7 is decreasing for
For every value of x, the function f(x) = `1/7^x` is ______
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
The function f(x) = tan-1 x is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
If f(x) = x + cosx – a then ______.
