1. **Problem 13:** Given the piecewise function $$g(x) = \begin{cases} ax + 3, & x \leq 1 \\ (x + a)^2 - 10, & x > 1 \end{cases}$$ Find the smaller integer value of $a$ such that $g(x)$ is continuous at $x=1$. 2. **Continuity condition:** For $g(x)$ to be continuous at $x=1$, the left-hand limit and right-hand limit at $x=1$ must be equal, and equal to $g(1)$: $$\lim_{x \to 1^-} g(x) = g(1) = \lim_{x \to 1^+} g(x)$$ 3. **Evaluate left-hand limit and $g(1)$:** Since $x \leq 1$, $$g(1) = a(1) + 3 = a + 3$$ 4. **Evaluate right-hand limit:** For $x > 1$, $$\lim_{x \to 1^+} g(x) = (1 + a)^2 - 10$$ 5. **Set continuity equation:** $$a + 3 = (1 + a)^2 - 10$$ 6. **Simplify:** $$a + 3 = (1 + a)^2 - 10$$ $$a + 3 = a^2 + 2a + 1 - 10$$ $$a + 3 = a^2 + 2a - 9$$ 7. **Bring all terms to one side:** $$0 = a^2 + 2a - 9 - a - 3$$ $$0 = a^2 + a - 12$$ 8. **Solve quadratic equation:** $$a^2 + a - 12 = 0$$ Use quadratic formula: $$a = \frac{-1 \pm \sqrt{1^2 - 4 \times 1 \times (-12)}}{2 \times 1} = \frac{-1 \pm \sqrt{1 + 48}}{2} = \frac{-1 \pm 7}{2}$$ 9. **Calculate roots:** $$a = \frac{-1 + 7}{2} = 3$$ $$a = \frac{-1 - 7}{2} = -4$$ 10. **Smaller integer value:** The smaller value of $a$ is $-4$. --- 11. **Problem 14:** Given $$y = \frac{12}{\sqrt[3]{2x + 1}}$$ Find the equation of the tangent line at the point $\left(\frac{7}{2}, 6\right)$ in the form $$4y + 21 = ?$$ 12. **Rewrite function:** $$y = 12 (2x + 1)^{-\frac{1}{3}}$$ 13. **Find derivative $y'$:** Using chain rule: $$y' = 12 \times -\frac{1}{3} (2x + 1)^{-\frac{4}{3}} \times 2 = -8 (2x + 1)^{-\frac{4}{3}}$$ 14. **Evaluate derivative at $x=\frac{7}{2}$:** Calculate inside the power: $$2 \times \frac{7}{2} + 1 = 7 + 1 = 8$$ So, $$y'\left(\frac{7}{2}\right) = -8 \times 8^{-\frac{4}{3}}$$ 15. **Simplify $8^{-\frac{4}{3}}$:** $$8 = 2^3$$ $$8^{-\frac{4}{3}} = (2^3)^{-\frac{4}{3}} = 2^{-4} = \frac{1}{16}$$ 16. **Calculate slope:** $$y'\left(\frac{7}{2}\right) = -8 \times \frac{1}{16} = -\frac{1}{2}$$ 17. **Equation of tangent line:** Using point-slope form: $$y - y_1 = m(x - x_1)$$ $$y - 6 = -\frac{1}{2} \left(x - \frac{7}{2}\right)$$ 18. **Multiply both sides by 4 to match form $4y + 21 = ?$:** $$4(y - 6) = 4 \times -\frac{1}{2} \left(x - \frac{7}{2}\right)$$ $$4y - 24 = -2x + 7$$ 19. **Add 21 to both sides:** $$4y - 24 + 21 = -2x + 7 + 21$$ $$4y - 3 = -2x + 28$$ 20. **Final form:** $$4y + 21 = -2x + 28 + 24 = -2x + 52$$ **Answer:** $$4y + 21 = -2x + 52$$