Advertisements
Advertisements
Question
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
Advertisements
Solution
f(x) = x3 – 12x2 – 144x + 13
∴ f'(x) = `d/dx(x^3 - 12x^2 - 144x + 13)`
= 3x2 – 12 x 2x – 144 x 1 + 0
= 3x2 – 24x – 144
= 3(x2 – 8x – 48)
(a) if is increasing if f'(x) ≥ 0
i.e. if 3(x2 – 8x – 48) ≥ 0
i.e. if x2 – 8x – 48 ≥ 0
i.e. if x2 – 8x ≥ 48
i.e. if x2 – 8x + 16 ≥ 48 + 16
i.e. if (x – 4)2 ≥ 64
i.e. if x – 4 ≥ 8 or x – 4 ≤ – 8
i.e. if x ≥ 12 or x ≤ – 4
∴ f is increasing if x ≤ – 4 or x ≥ 12,
i.e. x ∈ `( - oo, - 4] ∪ [12, oo)`.
(b) f is decreasing if f'(x) ≤ 0
i.e. if 3(x2 – 8x – 48) ≤ 0
i.e. if x2 – 8x – 48 ≤ 0
i.e. if x2 – 8x ≤ 48
i.e. if x2 – 8x + 16 ≤ 48 + 16
i.e. if (x – 4)2 ≤ 64
i.e. if – 8 ≤ x – 4 ≤ 8
i.e. if – 4 ≤ x ≤ 12
∴ f is decreasing if – 4 ≤ x ≤ 12, i.e. x ∈[– 4, 12].
APPEARS IN
RELATED QUESTIONS
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
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).
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
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) = 8 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
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 ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
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 ______
The function f(x) = sin x + 2x is ______
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
Which of the following functions is decreasing on `(0, pi/2)`?
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
Which of the following graph represent the strictly increasing function.
Function given by f(x) = sin x is strictly increasing in.
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
Find the value of x for which the function f(x)= 2x3 – 9x2 + 12x + 2 is decreasing.
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = `squarex^2 - square + square`
∴ f'(x) = `6(x - 1)(square)`
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < `square` and x > `square`
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > `square` and x < `square`
1 < `square` < 2
f(x) is decreasing if and only if x ∈ `square`
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
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 ______.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
y = log x satisfies for x > 1, the inequality ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
