Advertisements
Advertisements
प्रश्न
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
Advertisements
उत्तर
\[f\left( x \right) = x\left| x \right|, x \in R\]
\[\text { Case I: When x } \geq 0\]
\[f\left( x \right) = x\left| x \right| = x\left( x \right) = x^2 \]
\[ \Rightarrow f'\left( x \right) = 2x \geq 0 \forall x \geq 0\]
\[\text { So,} f\left( x \right)\text { is increasing for x } \geq 0 . \]
\[\text { Case II: When } x < 0\]
\[f\left( x \right) = x\left| x \right| = x\left( - x \right) = - x^2 \]
\[ \Rightarrow f'\left( x \right) = - 2x \geq 0 \forall x < 0\]
\[\text { So, }f\left( x \right)\text { is increasing for } x < 0 . \]
\[\text { Hence }, f\left( x \right)\text { is increasing for x } \in R . \]
APPEARS IN
संबंधित प्रश्न
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
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:
(x + 1)3 (x − 3)3
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.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Let f defined on [0, 1] be twice differentiable such that | f (x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Function f(x) = loga x is increasing on R, if
The function f(x) = x9 + 3x7 + 64 is increasing on
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
Show that f(x) = x – cos x is increasing for all x.
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
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] ______
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
For every value of x, the function f(x) = `1/7^x` is ______
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
The function `1/(1 + x^2)` is increasing in the interval ______
If f(x) = x3 – 15x2 + 84x – 17, then ______.
y = x(x – 3)2 decreases for the values of x given by : ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The function f (x) = 2 – 3 x is ____________.
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
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 ______.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
