Short Answer TypeFind the intervals in which the following functions are strictly increasing or decreasing:
6 – 9x – x 2
Find the intervals in which the following function is increasing or decreasing:
x3 – 6x2 + 9x + 15.
Find the intervals in which the following function f(x) is
(a) increasing (b) decreasing:
f (x) = 2x3 – 9x2 + 12x + 15
Determine the values of x for which the function f(x) = 2x3 – 24x + 5 is increasing or decreasing.
Find the intervals in which the functions f (x) = 2x3 – 15x2 + 36x + 1 is strictly increasing or decreasing. Also find the points on which the tangents are parallel to the x-axis.
f (x) = 2x3 –15x2 + 36x + 1
∴ f ' (x) = 6x2 – 30x + 36 = 6 (x2 – 5x + 6) = 6 (y – 2) (x – 3)
f ' (x) = 0 gives us 6 (r – 2) (x – 3) ⇒ x = 2, 3
The points x = 2, 3 divide the real line into three intervals (– ∞, 2), (2, 3), (3, ∞)
(i) In the interval (– ∞, 2), f ' (x) > 0
∴ f (x) is increasing in (2)
(ii) In the interval (2, 3), f ' (x) < 0
∴ f (x) is decreasing in (2, 3)
(iii) In the interval (3, ∞) f ' (x) > 0
∴ f (x) is increasing in (3, ∞)
∴ we see that f (x) increases in (– ∞, 2) ∪ (3, ∞) and decreasing in (2, 3).
Tangent is parallel to x-axis when f ' (x) = 0 i.e. 6 (x – 2) (x – 3) = 0 i.e. x = 2, 3
When x = 2, y = f (2) = 2 (2)3 – 15 (2)2 + 36 (2) + 1 – 16 –60 + 72 + 1 = 29
When x = 3, y = f (3) = 2(3)3 – 15 (3)2 + 36(3) + 1 = 54 – 135 + 108 + 1 = 28
∴ points are (2, 29), (3, 28).
Find the intervals in which the function
f(x) = x3 – 12x2 + 36x + 17 is
(a) strictly increasing (b) strictly decreasing
Find the intervals in which the function f (x) = 2x3 – 15x2 + 36x + 1 is
(a) strictly increasing (b) strictly decreasing
Find the intervals in which the following functions are strictly increasing or strictly decreasing:
x3 – 6x2 – 36x +4
Find the intervals in which the following functions are strictly increasing or strictly decreasing:
2x3 – 15x2 + 36x + 6
Find the intervals in which the following functions are strictly increasing or strictly decreasing:
6 + 12x + 3x2 – 2x3
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