Advertisements
Advertisements
प्रश्न
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Advertisements
उत्तर
\[f\left( x \right) = ax + b\]
\[\text { Let }x_1 , x_2 \text { in R such that } x_1 < x_2 . \]
\[\text { Then },\]
\[ x_1 < x_2 \]
\[ \Rightarrow a x_1 > a x_2 (\because a<0)\]
\[ \Rightarrow a x_1 + b > a x_2 + b\]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right)\]
\[\text { Thus }, x_1 < x_2 \]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right), \forall x_1 , x_2 \in R\]
\[\text { So },f\left( x \right) \text { is decreasing on R } .\]
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 least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Prove that the function f given by f(x) = log sin x is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi)`
The interval in which y = x2 e–x is increasing is ______.
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 5x3 − 15x2 − 120x + 3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
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(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) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
Find `dy/dx,if e^x+e^y=e^(x-y)`
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
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.
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
A circular pIate is contracting at the uniform rate of 5cm/sec. The rate at which the perimeter is decreasing when the radius of the circle is 10 cm Jong is
The function f(x) = x3 - 3x is ______.
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
If f(x) = x + cosx – a then ______.
y = log x satisfies for x > 1, the inequality ______.
Let f(x) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
