Advertisements
Advertisements
Question
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Advertisements
Solution
f(x) = x4 − 2x3 + 1
∴ f'(x) = 4x3 - 6x2 = 2x2 (2x - 3)
f(x) is a decreasing function, if f'(x) < 0
∴ 2x2 (2x - 3) < 0
As x2 is always positive,
(2x - 3) < 0
∴ 2x < 3
∴ x < `3/2`
Thus, f(x) is a decreasing function for x <`3/2`, i.e. `(- ∞, 3/2)`.
APPEARS IN
RELATED QUESTIONS
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).
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
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) ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
The function f(x) = cot−1 x + x increases in the interval
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
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).
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
The total cost function for production of articles is given as C = 100 + 600x – 3x2, then the values of x for which the total cost is decreasing is ______
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
The function f(x) = sin x + 2x is ______
The function f(x) = x2 – 2x is increasing in the interval ____________.
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:
Function given by f(x) = sin x is strictly increasing in.
Let 'a' be a real number such that the function f(x) = ax2 + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax2 – 6x + 15, x∈R has a ______.
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 ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Which statement about \[f'(x)=0\] and constant behaviour is correct?
