Advertisements
Advertisements
प्रश्न
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Advertisements
उत्तर
\[f\left( x \right) = x^3 - 15 x^2 + 75x - 50\]
\[f'\left( x \right) = 3 x^2 - 30x + 75\]
\[ = 3 \left( x^2 - 10x + 25 \right)\]
\[ = 3 \left( x - 5 \right)^2 > 0, \forall x \in R \left[ \because \text { Square of any function is always greater than zero } \right]\]
\[\text{ So,f(x)is an increasing function for all x} \in R.\]
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 intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
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(x) = (x − 1) (x − 2)2 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
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) = x3 − 6x2 + 9x + 15 ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
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 decreasing on R ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
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.
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
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 func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
Show that f(x) = x – cos x is increasing for all x.
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
The function f(x) = sin x + 2x is ______
y = x(x – 3)2 decreases for the values of x given by : ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
2x3 - 6x + 5 is an increasing function, if ____________.
Function given by f(x) = sin x is strictly increasing in.
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 + 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))` = ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
