Advertisements
Advertisements
प्रश्न
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).`
Advertisements
उत्तर
Here f(x) = log cos x
`therefore f'(x) = 1/(cos x) (- sin x) = - tan x`
(i) In the interval `(0, pi/2)`, tan x = + ve
∴ f' (x) = - ve
Hence, f is a decreasing function.
(ii) In the interval `(pi/2, pi)`, tan x = - ve
∴ f' (x) = - tan x = - ve
Hence, f is an increasing function.
APPEARS IN
संबंधित प्रश्न
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
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].
Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.
Find the intervals in which the function f given by `f(x) = (4sin x - 2x - x cos x)/(2 + cos x)` is (i) increasing (ii) decreasing.
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
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
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) = x2 + 2x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 12x2 + 18x + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
The slope of tangent at any point (a, b) is also called as ______.
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
The function f(x) = sin x + 2x is ______
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
y = x(x – 3)2 decreases for the values of x given by : ______.
Which of the following graph represent the strictly increasing function.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
The function f(x) = sin4x + cos4x is an increasing function if ______.
