Advertisements
Advertisements
Question
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
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 or 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, ∞)
APPEARS IN
RELATED QUESTIONS
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
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 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) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x2 + 2x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Let f defined on [0, 1] be twice differentiable such that | f (x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, 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.
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
If \[f'(x)>0\] throughout an interval, then \[f\] is
For \[f'(x)=12(x-3)(x+2)\], what are the critical points obtained from \[f'(x)=0\]?
