F(x) = (1/2)x⁴ - 4x + 1
To answer this question, we need to find the function 'F(x)' whose derivative is given by '2x³ - 4' and which passes through the point (2,1). The slope provided gives us the derivative of F(x):
F'(x) = 2x³ - 4
To find F(x), we need to integrate the derivative:
⇒ ∫F'(x) = ∫(2x³ - 4) dx
⇒ F(x) = ∫(2x³) dx - ∫(4) dx
⇒ F(x) = (2/4)x⁴ dx - 4x + C
So, the function is:
F(x) = (1/2)x⁴ - 4x + C
We know the function passes through the point (2,1). Plugging in x = 2 and F(x) = 1:
⇒ 1 = (1/2)(2)⁴ - 4(2) + C
⇒ 1 = (1/2)(16) - 8 + C
⇒ 1 = 8 - 8 + C
⇒ 1 = C
∴ C = 1
Therefore, the constant 'C' is 1, and the function is:
F(x) = (1/2)x4 - 4x + 1