Answer:
1. tangent slope is horizontal at (-5,197) and (1,-19)
2. the tangent line has slope -48 at points (-1, 37) and (-3, 141)
Explanation:
1. f(x) = 2x³+12x²-30x-3
f'(x) = 6x²+24x-30
the tangent is horizontal when f'(x) = 0
0 = 6x²+24x-30
0 = x²+4x-5
0 = (x+5)(x-1)
x= -5, 1
f(-5) = -250+300+150-3 = 197
f(1) = -19
tangent slope is horizontal at (-5,197) and (1,-19)
2. when f'(x) = -48
6x²+24x-30 = -48
6x²+24x+18 = 0
x²+4x+3 = 0
(x+1)(x+3) = 0
x = -1, -3
f(-1) = 37
f(-3)=141
the tangent line has slope -48 at points (-1, 37) and (-3, 141)