Advertisements
Advertisements
प्रश्न
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
Advertisements
उत्तर
\[\text { Here }, \]
\[f\left( x \right) = \left| x \right|\]
\[(a) \text { Let } x_1 , x_2 \in \left( 0, \infty \right) \text { such that } x_1 < x_2 . \text { Then },\]
\[ x_1 < x_2 \]
\[ \Rightarrow \left| x_1 \right| < \left| x_2 \right|\]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So },f\left( x \right) \text { is increasing on }\left( 0, \infty \right).\]
\[(b) \text { Let } x_1 , x_2 \in ( - \infty , 0]. \text { such that } x_1 < x_2 . \text { Then },\]
\[ x_1 < x_2 \]
\[ \Rightarrow \left| x_1 \right| > \left| x_2 \right|\]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) > f\left( x_2 \right), \forall x_1 , x_2 \in ( - \infty , 0].\]
\[\text { So },f\left( x \right) \text { is decreasing on }( - \infty , 0].\]
APPEARS IN
संबंधित प्रश्न
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Prove that the logarithmic function is strictly increasing on (0, ∞).
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 − 9x − x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 + 9x2 + 12x + 20 ?
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 ?
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
The function f(x) = xx decreases on the interval
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
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) = `x-(1)/x`, x ∈ R, x ≠ 0.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
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.
A ladder 20 ft Jong leans against a vertical wall. The top-end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is ______
The function f(x) = sin x + 2x is ______
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
The function f (x) = 2 – 3 x is ____________.
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).
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
