Question

given that the graph of (f(x)) passes through the point ((7, 7)) and that the slope of its tangent line at ((x, f(x))) is (6x + 5), what is (f(2))?

Explanation:

Find the general antiderivative of the slope function

$$ f'(x) = 6x + 5 $$

$$ f(x) = \int (6x + 5) \, dx = 3x^2 + 5x + C $$

Determine the constant of integration using the given point

$$ f(7) = 7 \implies 3(7)^2 + 5(7) + C = 7 $$

$$ 3(49) + 35 + C = 7 \implies 147 + 35 + C = 7 \implies 182 + C = 7 \implies C = -175 $$

$$ f(x) = 3x^2 + 5x - 175 $$

Evaluate the function at the target value

$$ f(2) = 3(2)^2 + 5(2) - 175 = 3(4) + 10 - 175 = 12 + 10 - 175 = -153 $$

Answer:

Given that the graph of \(f(x)\) passes through the point \((7, 7)\) and that the slope of its tangent line at \((x, f(x))\) is \(6x + 5\), what is \(f(2)\)? <blank>-153</blank>