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
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?
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 \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π) ?
Show that the function f given by f(x) = 10x is increasing for all x ?
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) ?
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.
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
Find `dy/dx,if e^x+e^y=e^(x-y)`
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) = x3 – 6x2 + 12x – 16, x ∈ R.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
The function f(x) = 9 - x5 - x7 is decreasing for
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
If f(x) = x + cosx – a then ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
The function f(x) = sin4x + cos4x is an increasing function if ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
