Advertisements
Advertisements
Question
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
Options
x < 2
x > 2
x > 3
1 < x < 2
Advertisements
Solution
1 < x < 2
\[f\left( x \right) = 2 x^3 - 9 x^2 + 12x + 29\]
\[f'\left( x \right) = 6 x^2 - 18x + 12\]
\[ = 6 \left( x^2 - 3x + 2 \right)\]
\[ = 6\left( x - 1 \right)\left( x - 2 \right)\]
\[\text { For f(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 6\left( x - 1 \right)\left( x - 2 \right) < 0 \]
\[ \Rightarrow \left( x - 1 \right)\left( x - 2 \right) < 0 \left[ \text { Since }6 > 0, 6\left( x - 1 \right)\left( x - 2 \right) < 0 \Rightarrow \left( x - 1 \right)\left( x - 2 \right) < 0 \right]\]
\[ \Rightarrow 1 < x < 2\]
\[\text { So,f(x) is decreasing for }1 < x < 2 .\]
APPEARS IN
RELATED QUESTIONS
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that the logarithmic function is strictly increasing on (0, ∞).
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
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 + 9x2 + 12x + 20 ?
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 \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R ?
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = loga x is increasing on R, if
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
Show that f(x) = x – cos x is increasing for all x.
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
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 function `1/(1 + x^2)` is increasing in the interval ______
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
Let 'a' be a real number such that the function f(x) = ax2 + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax2 – 6x + 15, x∈R has a ______.
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
The function f(x) = xex(1 − x), x ∈ R, is ______.
Let \[f\] be continuous on \[[a,b]\] and differentiable on \[(a,b)\]. If \[f'(x)\geq0\] for every \[x\in(a,b)\], what follows?
If \[f'(x)<0\] throughout an interval, then \[f\] is
After the domain has been divided into intervals, which quantity must be determined in each interval?
For \[f(x)=4x^3-6x^2-72x+30\], on which intervals is \[f\] increasing?
