Jika f''(x)=2x-3 dan f'(0)=f(0)=5, maka f(3) sama dengan

15.5

Pembahasan

Untuk mencari nilai f(3), kita perlu mengintegralkan f''(x) dua kali dan menggunakan informasi f'(0)=5 dan f(0)=5.

Mengintegralkan f''(x) = 2x - 3:
f'(x) = ∫(2x - 3) dx = x^2 - 3x + C1

Menggunakan f'(0) = 5:
5 = (0)^2 - 3(0) + C1
C1 = 5

Jadi, f'(x) = x^2 - 3x + 5.

Mengintegralkan f'(x) = x^2 - 3x + 5:
f(x) = ∫(x^2 - 3x + 5) dx = (1/3)x^3 - (3/2)x^2 + 5x + C2

Menggunakan f(0) = 5:
5 = (1/3)(0)^3 - (3/2)(0)^2 + 5(0) + C2
C2 = 5

Jadi, f(x) = (1/3)x^3 - (3/2)x^2 + 5x + 5.

Mencari f(3):
f(3) = (1/3)(3)^3 - (3/2)(3)^2 + 5(3) + 5
f(3) = (1/3)(27) - (3/2)(9) + 15 + 5
f(3) = 9 - 27/2 + 20
f(3) = 29 - 13.5
f(3) = 15.5