Advertisements
Advertisements
प्रश्न
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Advertisements
उत्तर
f(x) = 2x3 - 15x2 - 144x - 7
∴ f'(x) = 6x2 - 30x - 144
f(x) is an increasing function, if f'(x) > 0
∴ 6(x2 - 5x - 24) > 0
∴ 6(x + 3)(x - 8) > 0
∴ (x + 3)(x - 8) > 0
ab > 0 ⇔ a > 0 and b > 0 or a < 0 or b < 0
∴ Either (x + 3) > 0 and (x – 8) > 0 or
(x + 3) < 0 and (x – 8) < 0
Case 1: x + 3 > 0 and x - 8 > 0
∴ x > -3 and x > 8
∴ x > 8
Case 2: x + 3 < 0 and x - 8 < 0
∴ x < - 3 or x < 8
∴ x < - 3
Thus, f(x) is an increasing function for x < -3, or x > 8 i.e., (-∞, - 3) ∪ (8, ∞).
APPEARS IN
संबंधित प्रश्न
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 function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
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) = 5x3 − 15x2 − 120x + 3 ?
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
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) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π) ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
The function f(x) = x2 e−x is monotonic increasing when
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
Every invertible function is
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
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 f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R 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 ______
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is
If f(x) = x + cosx – a then ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
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 function f(x) = x3 + 3x is increasing in interval ______.
After the domain has been divided into intervals, which quantity must be determined in each interval?
