Advertisements
Advertisements
Question
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Advertisements
Solution
f(x) (x + 1)3 (x - 3)3
f'(x) = 3(x + 1)2 (x - 3)3 + (x + 1)3 (3 (x - 3)2)
= 6(x + 1)2 (x - 3)2 (x - 1)
f'(x) = 6(x + 1)2 (x - 3)2 (x - 1)
if, f'(x) = 0
6(x + 1)2 (x - 3)2 (x - 1) = 0
x = -1, 1, 3
x = -1, x = 1, x = 3 splits the real line into intervals `(- infty, -1), (-1, 1), (1,3)` and `(3, infty)`.
The function f is decreasing on the interval `(-infty, -1)`.
APPEARS IN
RELATED QUESTIONS
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
The interval in which y = x2 e–x is increasing is ______.
Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
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 − 5 ?
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
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] ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
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.
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
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.
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Show that f(x) = x – cos x is increasing for all x.
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically 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
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
The function `1/(1 + x^2)` is increasing in the interval ______
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Which of the following functions is decreasing on `(0, pi/2)`?
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
