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
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
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).
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(x) = 2x3 − 24x + 107 ?
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)2 ?
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(x) = x8 + 6x2 ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
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 decreasing on R ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Every invertible function is
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
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)
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
