The equation of a curve is \( y = f(x) \). The curve passes through the points \( (1, 3) \) and \( (3, 7) \). Given that \( f'(x) = 4x + p \), where \( p \) is a constant:

a) Find the value of \( p \).

b) Find the equation of the curve.

f ′ ( x ) = 4 x + p f ( x ) = 2 x 2 + p x + C 3 = 2 ⋅ 1 2 + p ⋅ 1 + C 7 = 2 ⋅ 3 2 + p ⋅ 3 + C 3 = 2 + p + C 7 = 18 + 3 p + C C = 1 − p 3 p = − 11 − C 3 p = − 11 − ( 1 − p ) 3 p = − 11 − 1 + p 2 p = − 12 p = − 6 C = 1 − ( − 6 ) C = 7 y = 2 x 2 − 6 x + 7

Answered by konrad509 | 2024-06-10

We found that the value of p is -6. Subsequently, the equation of the curve is given by f(x) = 2x² - 6x + 7, which passes through the points (1, 3) and (3, 7).
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Answered by konrad509 | 2024-12-20