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A)
f'(x) = 6x² – 4x + C
f'(-1) = 3
6(-1)² – 4(-1) + C = 3
6 + 4 + C = 3
C = -7
f'(x) = 6x² – 4x – 7
f(x) = 2x³ – 2x² – 7x + C
f(2) = 5
2(2)³ – 2(2)² – 7(2) + C = 5
16 – 8 – 14 + C = 5
-6 + C = 5
C = 11
f(x) = 2x³ – 2x² – 7x + 11
b) f'(x) = 4x³ + 6x + C
f'(1) = -2
4(1)³ + 6(1) + C = -2
4 + 6 + C = -2
10 + C = -2
C = -12
f'(x) = 4x³ + 6x – 12
f(x) = x⁴ + 3x² – 12x + C
f(-2) = 4
(-2)⁴ + 3(-2)² – 12(-2) + C = 4
16 + 12 + 24 + C = 4
52 + C = 4
C = -48
f(x) = x⁴ + 3x² – 12x – 48