Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\]
\[ = \frac{3 x^4 - 8 x^3 - 30 x^2 + 72x + 110}{10}\]
\[f'\left( x \right) = \frac{12 x^3 - 24 x^2 - 60x + 72}{10}\]
\[ = \frac{12}{10}\left( x^3 - 2 x^2 - 5x + 6 \right)\]
\[ = \frac{\left( x - 1 \right)\left( x^2 - x - 6 \right)}{10}\]
\[ = \frac{12}{10}\left( x - 1 \right)\left( x + 2 \right)\left( x - 3 \right)\]
\[\text { Here }, 1, 2 \text { and } 3 \text { are the critical points } . \]
\[\text { The possible intervals are }\left( - \infty - 2 \right),\left( - 2, 1 \right),\left( 1, 3 \right)\text { and }\left( 3, \infty \right).\]
\[\text { For }f(x)\text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \frac{12}{10}\left( x - 1 \right)\left( x + 2 \right)\left( x - 3 \right) > 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x + 2 \right)\left( x - 3 \right) > 0\]
\[ \Rightarrow x \in \left( - 2, 1 \right) \cup \left( 3, \infty \right)\]
\[\text { So },f(x)\text { is increasing on } x \in \left( - 2, 1 \right) \cup \left( 3, \infty \right) . \]
\[\text { For }f(x)\text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow \frac{12}{10}\left( x - 1 \right)\left( x + 2 \right)\left( x - 3 \right) < 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x + 2 \right)\left( x - 3 \right) < 0\]
\[ \Rightarrow x \in \left( - \infty - 2 \right) \cup \left( 1, 3 \right) \]
\[\text { So,}f(x)\text { is decreasing on } x \in \left( - \infty - 2 \right) \cup \left( 1, 3 \right) .\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Prove that the logarithmic function is strictly increasing on (0, ∞).
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing 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) = 2x3 − 12x2 + 18x + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
The function f(x) = cot−1 x + x increases in the interval
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
Every invertible function is
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
The slope of tangent at any point (a, b) is also called as ______.
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
The function f(x) = 9 - x5 - x7 is decreasing for
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
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 ______.
