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The graph of a function F(x) has a slope of 2x^3 - 4 at each point (x,y), and contains the point (2,1). Determine the function F(x).

Sagot :

Rochirion

Final answer:

F(x) = (1/2)x⁴ - 4x + 1

Explanation:

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

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