Advertisements
Advertisements
Question
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
Advertisements
Solution
f(x) = 2x3 – 15x2 – 144x – 7
∴ f'(x) = `"d"/("d"x)(2x^3 - 15x^2 - 144x - 7)`
= 2 × 3x2 – 15 × 2x – 144 × 1 – 0
= 6x2 – 30x – 144
= 6(x2 – 5x – 24)
(a) f is strictly increasing if f'(x) > 0
i.e., if 6(x2 – 5x – 24) > 0
i.e., if x2 – 5x –24 > 0
i.e., if x2 – 5x > 24
i.e., if `x^2 - 5x + (25)/(4) > 24 + (25)/(4)`
i.e., if `(x - 5/2)^2 > (121)/(4)`
i.e., if `x - (5)/(2) > (11)/(2) or x - (5)/(2) < - (11)/(2)`
i.e., if x > 8 or x < – 3
∴ f is strictly increasing if x < – 3 or x > 8.
(b) f is strictly decreasing if f''(x) < 0
i.e., if 6(x2 – 5x – 24) < 0
i.e., if x2 – 5x –24 < 0
i.e., if x2 – 5x < 24
i.e., if `x^2 - 5x + (25)/(4) < 24 + (25)/(4)`
i.e., if `(x - 5/2)^2 < (121)/(4)`
i.e., if `x - (5)/(2) < (11)/(2) or x - (5)/(2) > - (11)/(2)`
i.e., if `-(11)/(2) + (5)/(2) < x - (5)/(2) + (5)/(2) < (11)/(2) + (5)/(2)`
i.e., if – 3 < x < 8
∴ f is strictly decreasing if – 3 < x < 8.
RELATED QUESTIONS
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
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 − 12x2 + 18x + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?
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 f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
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] ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
The function f(x) = cot−1 x + x increases in the interval
The function f(x) = x2 e−x is monotonic increasing when
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Find `dy/dx,if e^x+e^y=e^(x-y)`
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Function given by f(x) = sin x is strictly increasing in.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
The function f(x) = x3 + 3x is increasing in interval ______.
The function f(x) = sin4x + cos4x is an increasing function if ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
