Final answer:
To find the function f'(x) and f(x), integrate the given derivative function with respect to x and find the constants. Substitute the given value of x to find f(1).
Step-by-step explanation:
To find the function f′(x), we can integrate the given derivative function f′′(x) = 7x + 2. Integrate it with respect to x to get f′(x) = ½x² + 2x + C, where C is a constant.
Given that f′(0) = 6, we can substitute x = 0 into the equation to find the value of C. 6 = ½(0)² + 2(0) + C. Solving this equation gives C = 6.
Therefore, the function f′(x) is f′(x) = ½x² + 2x + 6.
To find f(1), we can integrate f′(x) = ½x² + 2x + 6. Integrate it with respect to x to get f(x) = ½·ⁿ/₀ x³ + ⁹/₂ x² + 6x + D, where D is a constant.
Given that f(0) = 6, we can substitute x = 0 into the equation to find the value of D. 6 = ½(0)³ + ⁹/₂(0)² + 6(0) + D. Solving this equation gives D = 6.
Therefore, the function f(x) is f(x) = ½x³ + ⁹/₂ x² + 6x + 6.
To find f(1), substitute x = 1 into the equation f(x) = ½x³ + ⁹/₂ x² + 6x + 6 and solve. f(1) = ½(1)³ + ⁹/₂(1)² + 6(1) + 6 = 2.5 + 0.5 + 6 + 6 = 15.