Find all the points of local maxima and local minima of the function f(x) = (x – 1)³ (x + 1)² (a) 1, -1, -1/5 (b) 1, -1 (c) 1, -1/5 (d) -1, -1/5
Answers
Given : f(x) = (x – 1)³ (x + 1)²
To Find : local maxima and local minima of the function
a) 1, -1, -1/5 (b) 1, -1 (c) 1, -1/5 (d) -1, -1/5
Solution:
f(x) = (x – 1)³(x + 1)²
f'(x) = (x – 1)³ (2(x + 1)) + (x + 1)²(3(x - 1)²)
=> f'(x) = (x + 1)(x - 1)²( 2(x - 1) + 3(x + 1))
=> f'(x) = (x + 1)(x - 1)²( 2x - 2+ 3x + 3)
=> f'(x) = (x + 1)(x - 1)²( 5x + 1)
f'(x) = 0
=> x = - 1 , x = 1 , x = - 1/5
x = 1 is not local maxima or minima
as f(1) = 0 , f(0.9) < 0 and f(1.1) > 0
f(-1) = 0 , f(-0.9) < 0 , f(-1.1) < 0
Hence - 1 is local maxima
f(-1/5) = f(-0.2) = -1.106
f(-0.1) = -1.078
f(-0.3) = -1.076
Hence -1/5 is local minima
Local maxima and minima are -1 , - 1/5
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