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
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
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) = 5 + 36x + 3x2 − 2x3 ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
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 ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
Find `dy/dx,if e^x+e^y=e^(x-y)`
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).
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
Which of the following graph represent the strictly increasing function.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
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?
The function \[f(x)=x^3-3x^2+4x\], \[x\in\mathbf{R}\], is
