given that the graph of ( f ) passes through the point ( (3,9) ) and that the slope of its tangent line at (…

Explanation:

Step1: Find the antiderivative of the slope function

The slope of the tangent line is (f^{\prime}(x)=7 - 6x). The antiderivative of (f^{\prime}(x)) is (f(x)=\int(7 - 6x)dx). Using the power rule (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)) and (\int a dx=ax + C) ((a) is a constant), we have: (f(x)=7x-6\times\frac{x^{2}}{2}+C=7x - 3x^{2}+C).

Step2: Use the given point to find the value of (C)

Since the graph of (f) passes through the point ((3,9)), substitute (x = 3) and (f(3)=9) into (f(x)=7x-3x^{2}+C). (9=7\times3-3\times3^{2}+C). (9 = 21-27 + C). (9=-6 + C), so (C=15).

Step3: Find (f(1))

Substitute (x = 1) and (C = 15) into (f(x)=7x-3x^{2}+C). (f(1)=7\times1-3\times1^{2}+15). (f(1)=7 - 3+15). (f(1)=19).

Answer:

(19)