Advertisements
Advertisements
Question
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
Advertisements
Solution
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = 8 + 36x + 3 x^2 - 2 x^3 \]
\[f'\left( x \right) = 36 + 6x - 6 x^2 \]
\[ = - 6 \left( x^2 - x - 6 \right)\]
\[ = - 6 \left( x - 3 \right)\left( x + 2 \right)\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow - 6 \left( x - 3 \right)\left( x + 2 \right) > 0 \]
\[ \Rightarrow \left( x - 3 \right)\left( x + 2 \right) < 0 \left[ \text { Since } - 6 < 0, - 6 \left( x - 3 \right)\left( x + 2 \right) > 0 \Rightarrow \left( x - 3 \right)\left( x + 2 \right) < 0 \right]\]
\[ \Rightarrow - 2 < x < 3\]
\[ \Rightarrow x \in \left( - 2, 3 \right)\]
\[\text { So,}f(x)\text { is increasing on} \left( - 2, 3 \right) . \]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow - 6 \left( x - 3 \right)\left( x + 2 \right) < 0\]
\[ \Rightarrow \left( x - 3 \right)\left( x + 2 \right) > 0 \left[ \text { Since } - 6 < 0, - 6 \left( x - 3 \right)\left( x + 2 \right) < 0 \Rightarrow \left( x - 3 \right)\left( x + 2 \right) > 0 \right]\]
\[ \Rightarrow x < - 2 \ or \ x > 3 \]
\[ \Rightarrow x \in \left( - \infty , - 2 \right) \cup \left( 3, \infty \right)\]
\[\text { So },f(x)\text { is decreasing on } \left( - \infty , - 2 \right) \cup \left( 3, \infty \right) .\]
APPEARS IN
RELATED QUESTIONS
Prove that the logarithmic function is strictly increasing on (0, ∞).
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2 ?
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) = 2x3 − 24x + 107 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
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 the function f(x) = ax + b is decreasing for all x ∈ R ?
The function f(x) = xx decreases on the interval
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
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).
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
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
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
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) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
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.
The function f(x) = sin4x + cos4x is an increasing function if ______.
The function f(x) = xex(1 − x), x ∈ R, is ______.
