Question

求解题过程，感谢🙏

Answer 1

(1)
f(x)=2x^3+3mx^2-36x+m
f'(x)=6x^2+6mx-36
f'(-1)=-36
6-6m-36=-36
m=1
(2)
m=1

f'(x)
=6x^2+6mx-36
=6x^2+6x-36
f'(x)=0
6x^2+6x-36=0
x^2-x-6=0
(x+2)(x-3)=0
x=-2 or 3
f''(x) = 12x+6
f''(-2) =-24+6=-18 <0 (max)
f''(3) =36+6=42 >0 (min)
f(x)=2x^3+3x^2-36x+1
max f(x) =f(-2) =-16+12+72+1 =69
min f(x) = f(3) = 54+27-108+1 =-26

Answer 2

(1) m=1

(2)单调递增（-∞,-3）,(2,+∞)

单调递减[-3,2]

Answer 3

（1）解：f(x)=2x^3+3mx^2-36x+m
f'(x)=6x^2+6mx-36
f'(-1)=-36
6-6m-36=-36
m=1
（2）：m=1
f(x) = 2x³ +3x² -36x + 1
f'(x) = 6x² + 6x - 36 = 6 (x + 3)(x - 2)
f'(x) = 6 (x + 3)(x - 2) ≤ 0 ,
-3 ≤ x ≤ 2 , f(x) 单调递减
f'(x) = 6 (x + 3)(x - 2) ≥ 0 ,
x ≤ -3 或 x ≥ 2 , f(x) 单调递增