Advertisements
Advertisements
Question
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Advertisements
Solution 1
\[f\left( x \right) = \sin x - \cos x\]
\[f'\left( x \right) = \cos x + \sin x\]
\[ = \cos x\left( 1 + \frac{\sin x}{\cos x} \right)\]
\[ = \cos x\left( 1 + \cot x \right)\]
\[\text { Here, } \]
\[\frac{- \pi}{4} < x < \frac{\pi}{4}\]
\[ \Rightarrow \cos x > 0 . . . \left( 1 \right)\]
\[\text { Also, } \]
\[\frac{- \pi}{4} < x < \frac{\pi}{4} \Rightarrow - 1 < \cot x < 1\]
\[ \Rightarrow 0 < 1 + \cot x < 2\]
\[ \Rightarrow 1 + \cot x > 0 . . . \left( 2 \right)\]
\[\cos x\left( 1 + \cot x \right) > 0, \forall x \in \left( \frac{- \pi}{4}, \frac{\pi}{4} \right) \left[ \text { From eqs }. (1) \text { and }(2) \right]\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in \left( \frac{- \pi}{4}, \frac{\pi}{4} \right)\]
\[\text { So,}f\left( x \right) \text { is increasing on }\left( \frac{- \pi}{4}, \frac{\pi}{4} \right).\]
Solution 2
f(x) = sinx − cosx
We differentiate f(x) with respect to x:
`f'(x) = d/dx (sinx-cosx) = cosx + sin x`
f′(x) = cosx + sinx
I `(-pi/4, pi/4)`, both sinx and cosx are positive.
Therefore, f′(x) = cosx + sinx > 0 throughout that interval.
This implies that f(x) is strictly increasing on `(-pi/4, pi/4)`
f(x) = sinx − cosx is an increasing function on `(-pi/4, pi/4)`
APPEARS IN
RELATED QUESTIONS
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 + 9x2 + 12x + 20 ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?
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) = 2x3 − 24x + 7 ?
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 ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
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) = x2 − x sin x is an increasing function on (0, π/2) ?
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
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
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
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)`.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
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:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
Show that function f(x) = tan x is increasing in `(0, π/2)`.
Let 'a' be a real number such that the function f(x) = ax2 + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax2 – 6x + 15, x∈R has a ______.
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
Let f(x) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
