Advertisements
Advertisements
प्रश्न
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
Advertisements
उत्तर
\[\text { Here }, \]
\[f\left( x \right) = ax + b\]
\[\text { Let } x_1 , x_2 \text { in R such that } x_1 < x_2 . \text { Then },\]
\[ x_1 < x_2 \]
\[ \Rightarrow a x_1 < a x_2 \left[ \because a>0 \right]\]
\[ \Rightarrow a x_1 + b < a x_2 + b\]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]
\[ \therefore x_1 < x_2 \]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in R \]
\[\text { So },f\left( x \right) \text { is increasing on R } .\]
APPEARS IN
संबंधित प्रश्न
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Prove that the logarithmic function is strictly increasing on (0, ∞).
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
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) = (x − 1) (x − 2)2 ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Function f(x) = ax is increasing on R, if
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.
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 decreasing : f(x) = x3 – 9x2 + 24x + 12
Show that f(x) = x – cos x is increasing for all x.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
The slope of tangent at any point (a, b) is also called as ______.
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
Function given by f(x) = sin x is strictly increasing in.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
A function \[f\] is monotonic in an interval when it is which of the following?
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] decreasing at \[x_0\]?
After the domain has been divided into intervals, which quantity must be determined in each interval?
For \[f(x)=4x^3-6x^2-72x+30\], on which intervals is \[f\] increasing?
Which statement about \[f'(x)=0\] and constant behaviour is correct?
