Raj is playing with a spring by throwing it in the air which is moving the function
f(x) = 3x4 −4x3 −12x2 + 5
17. Find the critical point of x when it touches the x-axis : a) 0,1,2 b)0,-1,2 c)0,1,-2 d)1,2,-2
18. Find the interval in which the spring is strictly increasing. a)(−1,0) ∪ (2,∞) b)(1,0) ∪ (2,∞) c)(1,∞) 19. Find the interval in which the spring is strictly decreasing.
a) (1,0) ∪ (2,∞) b)(−∞,−1) ∪ (0, 2) c)(−∞,2) 20. Find the values of x at which spring has local maxima
a) 2 b) -1 c) 1 d) 0
21. What is the maximum height caused by spring?
a) 10 b)72 c)5 d)0
Answers
Given : f(x) = 3x⁴ −4x³ −12x² + 5
To Find : critical point of x
a) 0,1,2 b)0,-1,2 c)0,1,-2 d)1,2,-2
the interval in which the spring is strictly increasing
a)(−1,0) ∪ (2,∞) b)(1,0) ∪ (2,∞) c)(1,∞) 19.
interval in which the spring is strictly decreasing.
a) (1,0) ∪ (2,∞) b)(−∞,−1) ∪ (0, 2) c)(−∞,2)
Solution:
f(x) = 3x⁴ −4x³ −12x² + 5
f'(x) = 12x³ −12x² -24x
= 12x(x² - x - 2)
= 12x(x - 2)(x + 1)
f"(x) = 0
=> x = 0 , 2 , - 1
critical point of x = 0 , - 1 , 2
intervals (−∞,−1) , (−1,0) , (0, 2) (2,∞)
interval f'(x) = 12x(x - 2)(x + 1)
(−∞,−1) (-)(-)(-) = - < 0 strictly decreasing
(−1,0) (-)(-)(+) = + > 0 strictly increasing
(0, 2) (+)(-)(+) = - < 0 strictly decreasing
(2,∞) (+)(+)(+) = + > 0 strictly increasing
strictly increasing (−1,0) U (2,∞)
strictly decreasing (−∞,−1) U (0, 2)
f''(x) = 36x² -24x - 24
x = 0 => f''(x) = -24 < 0 hence local maxima
x = -1 => f''(x) = 36 + 24 - 24 = 36 > 0 hence local minima
x =2 => f''(x) = 144 - 48 - 24 = 72 > 0 hence local minima
local maxima = 0
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