Advertisements
Advertisements
प्रश्न
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
उत्तर
f(x) = 2x3 - 15x2 + 36x + 1
∴ f'(x) = 6x2 - 30x + 36
= 6(x2 - 5x + 6)
= 6(x - 3)(x - 2)
f(x) is an increasing function, if f'(x) > 0
∴ 6(x - 3)(x - 2) > 0
∴ (x - 3)(x - 2) > 0
ab > 0 ⇔ a > 0 and b > 0 or a < 0 or b < 0
∴ Either (x – 3) > 0 and (x – 2) > 0 or
(x – 3) < 0 and (x – 2) < 0
Case 1: x – 3 > 0 and x – 2 > 0
∴ x > 3 and x > 2
∴ x > 3
Case 2: x – 3 < 0 and x – 2 < 0
∴ x < 3 and x < 2
∴ x < 2
Thus, f(x) is an increasing function for x < 2 or x > 3, i.e., (- ∞, 2) ∪ (3, ∞)
APPEARS IN
संबंधित प्रश्न
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
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
Prove that the function f given by f(x) = log sin x is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi)`
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Show that f(x) = e2x is increasing on R.
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
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).
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function 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 ?
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
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).
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
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 ______
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
Which of the following functions is decreasing on `(0, pi/2)`?
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Which of the following graph represent the strictly increasing function.
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
