Advertisements
Advertisements
प्रश्न
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).
Advertisements
उत्तर
\[Here, \]
\[f\left( x \right) = \cos x\]
\[\text{Domain of cos x is}\left( - \pi, \pi \right).\]
\[ \Rightarrow f'\left( x \right) = - \sin x\]
\[\text{For x} \in \left( - \pi, 0 \right), \sin x < 0 \left[ \because \text{sine function is negative in third and fourth quadrant }\right]\]
\[ \Rightarrow - \sin x > 0\]
\[ \Rightarrow f'\left( x \right) > 0\]
\[So, \text{cos x is increasing in} \left( - \pi, 0 \right) . \]
\[\text{For x} \in \left( 0, \pi \right)),\sin x > 0 \left[ \because \text{sine function is positive in first and second quadrant }\right]\]
\[ \Rightarrow - \sin x < 0\]
\[ \Rightarrow f'\left( x \right) < 0\]
\[\text{So,f(x) is decreasing on}\left( 0, \pi \right).\]
\[\text{Thus,f(x) is neither increasing nor decreasing in}\left( - \pi, \pi \right).\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Prove that the function f given by f(x) = log cos x is strictly decreasing on `(0, pi/2)` and strictly increasing on `((3pi)/2, 2pi).`
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 an increasing function on R ?
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) = x3 − 6x2 − 36x + 2 ?
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
Let f defined on [0, 1] be twice differentiable such that | f (x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
A function \[f\] is monotonic in an interval when it is which of the following?
For \[f(x)=x^3-3x^2+4x\], what is \[f'(x)\]?
For \[f'(x)=12(x-3)(x+2)\], what are the critical points obtained from \[f'(x)=0\]?
