Advertisements
Advertisements
प्रश्न
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Advertisements
उत्तर
\[f\left( x \right) = e^\frac{1}{x} \]
\[f'\left( x \right) = e^\frac{1}{x} \frac{d}{dx}\left( \frac{1}{x} \right)\]
\[ = e^\frac{1}{x} \left( \frac{- 1}{x^2} \right)\]
\[ = - \frac{e^\frac{1}{x}}{x^2}\]
\[\text { Here, }e^\frac{1}{x} > 0 \text { and } x^2 > 0, \text { for any real value of} x \neq 0.\]
\[\therefore f \left( x \right) = - \frac{e^\frac{1}{x}}{x^2} < 0, \forall x \in R, x \neq 0\]
\[\text { So,f(x) is a decreasing function }.\]
APPEARS IN
संबंधित प्रश्न
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 ?
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 f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
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 − 24x + 107 ?
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 \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
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.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
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`
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) 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 values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
The function f(x) = tanx – x ______.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
The function f(x) = sin4x + cos4x is an increasing function if ______.
Let \[f\] be continuous on \[[a,b]\] and differentiable on \[(a,b)\]. If \[f'(x)\geq0\] for every \[x\in(a,b)\], what follows?
If \[f'(x)>0\] throughout an interval, then \[f\] is
If \[f'(x)<0\] throughout an interval, then \[f\] is
The function \[f(x)=x^3-3x^2+4x\], \[x\in\mathbf{R}\], is
Which statement about \[f'(x)=0\] and constant behaviour is correct?
