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Application Of Derivatives

Question
CBSEENMA12035308
Wired Faculty App

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.

Solution

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).

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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