Advertisements
Advertisements
प्रश्न
Show that f(x) = e2x is increasing on R.
Show that the function given by f (x) = e 2x is increasing on R.
Advertisements
उत्तर १
\[f\left( x \right) = e^{2x} \]
\[f'\left( x \right) = 2 e^{2x} \]
\[\text { Now,} \]
\[x \in R\]
Since the value of `e^{2x}` text is always positive for any real value of x, ` e^{2x}` > 0 .
\[ \Rightarrow 2 e^{2x} > 0\]
\[ \Rightarrow f'\left( x \right) > 0\]
\[\text { So,f(x)is increasing on R} .\]
उत्तर २
We have f(x) = e2x
f'(x) = 2e2x > 0, x `in` R
f is strictly increasing on R
APPEARS IN
संबंधित प्रश्न
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
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.
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 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
Function f(x) = | x | − | x − 1 | is monotonically increasing when
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
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.
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
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
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
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 ______
The function f(x) = sin x + 2x is ______
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
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`
If f(x) = x + cosx – a then ______.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
