1. **State the problem:** Find the equation of the tangent line to the function $$f(x) = (1 + 12\sqrt{x})(4 - x^2)$$ at $$x = 9$$ and fill in the blank in the equation $$y = -2849 - \_\_\_ (x - 9)$$. 2. **Recall the formula for the tangent line:** The tangent line to $$f(x)$$ at $$x = a$$ is given by: $$ y = f(a) + f'(a)(x - a) $$ where $$f'(a)$$ is the derivative of $$f(x)$$ evaluated at $$x = a$$. 3. **Find $$f(9)$$:** Calculate $$f(9) = (1 + 12\sqrt{9})(4 - 9^2) = (1 + 12 \times 3)(4 - 81) = (1 + 36)(-77) = 37 \times (-77) = -2849$$. 4. **Find $$f'(x)$$:** Use the product rule: $$f(x) = u(x) v(x)$$ where $$u(x) = 1 + 12\sqrt{x}$$ and $$v(x) = 4 - x^2$$. - Derivative of $$u(x)$$: $$ u u'(x) = 12 \times \frac{1}{2\sqrt{x}} = \frac{6}{\sqrt{x}}$$ - Derivative of $$v(x)$$: $$v'(x) = -2x$$ - Apply product rule: $$ f'(x) = u'(x) v(x) + u(x) v'(x) = \frac{6}{\sqrt{x}} (4 - x^2) + (1 + 12\sqrt{x})(-2x) $$ 5. **Evaluate $$f'(9)$$:** Calculate each term: - $$\frac{6}{\sqrt{9}} (4 - 81) = \frac{6}{3} \times (-77) = 2 \times (-77) = -154$$ - $$ (1 + 12 \times 3)(-2 \times 9) = 37 \times (-18) = -666$$ Sum: $$f'(9) = -154 - 666 = -820$$ 6. **Write the tangent line equation:** $$y = f(9) + f'(9)(x - 9) = -2849 - 820 (x - 9)$$ **Final answer:** The blank is $$820$$.