Advertisements
Advertisements
Question
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Advertisements
Solution
f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = 6x2 – 30x – 84
= 6(x2 – 5x – 14)
= 6(x2 – 7x + 2x – 14)
= 6(x – 7)(x + 2)
f(x) is an decreasing function, if f'(x) < 0
∴ 6(x – 7)(x + 2) < 0
∴ (x – 7)(x + 2) < 0
ab < 0 `⇔` a > 0 and b < 0 or a < 0 or b > 0
∴ Either (x – 7) > 0 and (x + 2) < 0 or
(x – 7) < 0 and (x + 2) > 0
Case 1: x – 7 > 0 and x + 2 < 0
∴ x > 7 and x < –2, which is not possible.
Case 2: x – 7 < 0 and x + 2 > 0
∴ x < 7 and x > –2
Thus, f(x) is an decreasing function for –2 < x < 7 i.e., (–2, 7)
APPEARS IN
RELATED QUESTIONS
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Prove that the logarithmic function is strictly increasing on (0, ∞).
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
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
Function f(x) = loga x is increasing on R, if
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The function f(x) = x2 – 2x is increasing in the interval ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
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) = `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 ______.
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] increasing at \[x_0\]?
As one moves from left to right on a graph, what does an increase in the \[y\]-values indicate?
