Advertisements
Advertisements
प्रश्न
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Advertisements
उत्तर
f(x) = `x + (25)/x`
∴ f'(x) = `d/dx(x + 25/x)`
= 1 + 25 (– 1)x–2
= `1 - (25)/x^2`
f is strictly decreasing if f'(x) < 0
i.e. if `1 - (25)/x^2 < 0`
i.e. if `1 < (25)/x^2`
i.e. if x2 < 25
i.e. if –5 < x < 5, x ≠ 0
i.e. if x ∈ (– 5, 5) – { 0 }
∴ f is strictly decreasing if x ∈ (– 5, 5) – { 0 }.
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 ?
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Prove that the logarithmic function is strictly increasing on (0, ∞).
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function 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) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
Every invertible function is
Function f(x) = ax is increasing on R, if
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Find `dy/dx,if e^x+e^y=e^(x-y)`
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
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) = x4 − 2x3 + 1
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
The total cost function for production of articles is given as C = 100 + 600x – 3x2, then the values of x for which the total cost is decreasing is ______
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
The function f(x) = tanx – x ______.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
