Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = x^4 - 4x\]
\[f'\left( x \right) = 4 x^3 - 4\]
\[ = 4\left( x^3 - 1 \right)\]
\[\text { For}f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 4\left( x^3 - 1 \right) > 0 \]
\[ \Rightarrow x^3 - 1 > 0\]
\[ \Rightarrow x^3 > 1\]
\[ \Rightarrow x > 1\]
\[ \Rightarrow x \in \left( 1, \infty \right)\]
\[\text { So,}f(x)\text { is increasing on }\left( 1, \infty \right) . \]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 4\left( x^3 - 1 \right) < 0\]
\[ \Rightarrow x^3 - 1 < 0\]
\[ \Rightarrow x^3 < 1\]
\[ \Rightarrow x < 1\]
\[ \Rightarrow x \in \left( - \infty , 1 \right)\]
\[\text { So,}f(x)\text { is decreasing on }\left( - \infty , 1 \right).\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (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) = 5 + 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 − 15x2 + 36x + 1 ?
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) = x8 + 6x2 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = e2x is increasing on R.
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
What are the values of 'a' for which f(x) = ax is increasing on R ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
Function f(x) = cos x − 2 λ x is monotonic decreasing when
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The function f (x) = x2, for all real x, is ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
The function \[f(x)=x^3-3x^2+4x\], \[x\in\mathbf{R}\], is
For \[f(x)=4x^3-6x^2-72x+30\], on which intervals is \[f\] increasing?
