Advertisements
Advertisements
Question
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
Solution
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\]
\[ = \frac{3 x^4 - 8 x^3 - 30 x^2 + 72x + 110}{10}\]
\[f'\left( x \right) = \frac{12 x^3 - 24 x^2 - 60x + 72}{10}\]
\[ = \frac{12}{10}\left( x^3 - 2 x^2 - 5x + 6 \right)\]
\[ = \frac{\left( x - 1 \right)\left( x^2 - x - 6 \right)}{10}\]
\[ = \frac{12}{10}\left( x - 1 \right)\left( x + 2 \right)\left( x - 3 \right)\]
\[\text { Here }, 1, 2 \text { and } 3 \text { are the critical points } . \]
\[\text { The possible intervals are }\left( - \infty - 2 \right),\left( - 2, 1 \right),\left( 1, 3 \right)\text { and }\left( 3, \infty \right).\]
\[\text { For }f(x)\text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \frac{12}{10}\left( x - 1 \right)\left( x + 2 \right)\left( x - 3 \right) > 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x + 2 \right)\left( x - 3 \right) > 0\]
\[ \Rightarrow x \in \left( - 2, 1 \right) \cup \left( 3, \infty \right)\]
\[\text { So },f(x)\text { is increasing on } x \in \left( - 2, 1 \right) \cup \left( 3, \infty \right) . \]
\[\text { For }f(x)\text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow \frac{12}{10}\left( x - 1 \right)\left( x + 2 \right)\left( x - 3 \right) < 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x + 2 \right)\left( x - 3 \right) < 0\]
\[ \Rightarrow x \in \left( - \infty - 2 \right) \cup \left( 1, 3 \right) \]
\[\text { So,}f(x)\text { is decreasing on } x \in \left( - \infty - 2 \right) \cup \left( 1, 3 \right) .\]
APPEARS IN
RELATED QUESTIONS
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
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)`
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 − 24x + 107 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
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 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
The function f(x) = x9 + 3x7 + 64 is increasing on
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 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) = 2x3 - 15x2 - 144x - 7
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
The function f(x) = 9 - x5 - x7 is decreasing for
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 ______
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
The function f (x) = 2 – 3 x is ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
