Advertisements
Advertisements
Question
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
Advertisements
Solution
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 and 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, ∞)
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 function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
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)
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Prove that the logarithmic function is strictly increasing on (0, ∞).
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R ?
The function f(x) = cot−1 x + x increases in the interval
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Function f(x) = ax is increasing on R, if
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
Which of the following functions is decreasing on `(0, pi/2)`?
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
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) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.
The function f(x) = x3 + 3x is increasing in interval ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
If \[f'(x)>0\] throughout an interval, then \[f\] is
As one moves from left to right on a graph, what does an increase in the \[y\]-values indicate?
