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
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
The interval in which y = x2 e–x is increasing is ______.
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).
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
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) = x8 + 6x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
The slope of tangent at any point (a, b) is also called as ______.
The function f(x) = 9 - x5 - x7 is decreasing for
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
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`
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] decreasing at \[x_0\]?
Let \[f\] be continuous on \[[a,b]\] and differentiable on \[(a,b)\]. If \[f'(x)\leq0\] for every \[x\in(a,b)\], what follows?
As one moves from left to right on a graph, what does a decrease in the \[y\]-values indicate?
As one moves from left to right on a graph, what do constant \[y\]-values indicate?
For \[f(x)=x^3-3x^2+4x\], what is \[f'(x)\]?
For \[f(x)=4x^3-6x^2-72x+30\], which factorization of \[f'(x)\] is correct?
