Advertisements
Advertisements
Question
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
Advertisements
Solution
\[\text { Here }, \]
\[f\left( x \right) = \left| x \right|\]
\[(a) \text { Let } x_1 , x_2 \in \left( 0, \infty \right) \text { such that } x_1 < x_2 . \text { Then },\]
\[ x_1 < x_2 \]
\[ \Rightarrow \left| x_1 \right| < \left| x_2 \right|\]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So },f\left( x \right) \text { is increasing on }\left( 0, \infty \right).\]
\[(b) \text { Let } x_1 , x_2 \in ( - \infty , 0]. \text { such that } x_1 < x_2 . \text { Then },\]
\[ x_1 < x_2 \]
\[ \Rightarrow \left| x_1 \right| > \left| x_2 \right|\]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) > f\left( x_2 \right), \forall x_1 , x_2 \in ( - \infty , 0].\]
\[\text { So },f\left( x \right) \text { is decreasing on }( - \infty , 0].\]
APPEARS IN
RELATED QUESTIONS
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.
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.
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)`
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 ?
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
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 ?
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
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) ?
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 interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
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
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
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 f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).
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.
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
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) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
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).
Find the value of x for which the function f(x)= 2x3 – 9x2 + 12x + 2 is decreasing.
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = `squarex^2 - square + square`
∴ f'(x) = `6(x - 1)(square)`
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < `square` and x > `square`
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > `square` and x < `square`
1 < `square` < 2
f(x) is decreasing if and only if x ∈ `square`
