Advertisements
Advertisements
Question
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
Advertisements
Solution
f(x) = x3 – 6x2 – 36x + 7
∴ f′(x) = 3x2 – 12x – 36
= 3(x2 – 4x – 12)
= 3(x – 6)(x + 2)
f(x) is strictly increasing, if f′(x) > 0
∴ 3(x – 6)(x + 2) > 0
∴ (x – 6)(x + 2) > 0
ab > 0 ⇔ a > 0 and b > 0 or a < 0 and b < 0
Either x – 6 > 0 and x + 2 > 0
or
x – 6 < 0 and x + 2 < 0
Case I: x – 6 > 0 and x + 2 > 0
∴ x > 6 and x > – 2
∴ x > 6
Case II: x – 6 < 0 and x + 2 < 0
∴ x < 6 and x < – 2
∴ x < – 2
Thus, f(x) is strictly increasing for x ∈ (−∞ −2) ∪ (6, ∞).
APPEARS IN
RELATED QUESTIONS
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Prove that the logarithmic function is strictly increasing on (0, ∞).
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
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
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 12x2 + 18x + 15 ?
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) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Function f(x) = loga x is increasing on R, if
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
The function f(x) = x3 - 3x is ______.
The function f(x) = sin x + 2x is ______
The function `1/(1 + x^2)` is increasing in the interval ______
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
Which of the following graph represent the strictly increasing function.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
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 ______.
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
y = log x satisfies for x > 1, the inequality ______.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] decreasing at \[x_0\]?
For \[f'(x)=12(x-3)(x+2)\], what is the nature of \[f\] on \[(-2,3)\]?
