Advertisements
Advertisements
प्रश्न
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4) ?
उत्तर
\[f\left( x \right) = \sin x - \cos x\]
\[f'\left( x \right) = \cos x + \sin x\]
\[ = \cos x\left( 1 + \frac{\sin x}{\cos x} \right)\]
\[ = \cos x\left( 1 + \cot x \right)\]
\[\text { Here, } \]
\[\frac{- \pi}{4} < x < \frac{\pi}{4}\]
\[ \Rightarrow \cos x > 0 . . . \left( 1 \right)\]
\[\text { Also, } \]
\[\frac{- \pi}{4} < x < \frac{\pi}{4} \Rightarrow - 1 < \cot x < 1\]
\[ \Rightarrow 0 < 1 + \cot x < 2\]
\[ \Rightarrow 1 + \cot x > 0 . . . \left( 2 \right)\]
\[\cos x\left( 1 + \cot x \right) > 0, \forall x \in \left( \frac{- \pi}{4}, \frac{\pi}{4} \right) \left[ \text { From eqs }. (1) \text { and }(2) \right]\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in \left( \frac{- \pi}{4}, \frac{\pi}{4} \right)\]
\[\text { So,}f\left( x \right) \text { is increasing on }\left( \frac{- \pi}{4}, \frac{\pi}{4} \right).\]
APPEARS IN
संबंधित प्रश्न
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).
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
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) ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
The function f(x) = cot−1 x + x increases in the interval
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
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. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
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).
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure Ec of a person with income I is given as Ec = ( 0.0003 ) I2 + ( 0.075 ) I when I = 1000.
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
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.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
The slope of tangent at any point (a, b) is also called as ______.
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
In case of decreasing functions, slope of tangent and hence derivative is ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
Which of the following graph represent the strictly increasing function.
Function given by f(x) = sin x is strictly increasing in.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
A function f is said to be increasing at a point c if ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
