1. The problem asks to find the equation of the tangent line to the curve $$y = x^{3/2} + 3x + 1$$ at the given point $(2,3)$. 2. First, verify the point lies on the curve by substituting $x=2$: $$y = 2^{3/2} + 3(2) + 1 = \\ 2^{1.5} + 6 + 1 = \\ \sqrt{2^3} + 7 = \\ \sqrt{8} + 7 = \\ 2\sqrt{2} + 7 \approx 2.828 + 7 = 9.828.$$ The point $(2,3)$ does not satisfy $y=x^{3/2} + 3x + 1$. This means either the point or the function is misstated. 3. Assuming the original function intended is $y = x^{3/2} - 3x + 1$ to suit the point $(2,3)$, verify again: $$y = 2^{3/2} - 3(2) + 1 = 2.828 - 6 + 1 = -2.172$$ which also does not equal 3. 4. Without changing the function, but since the user provided the point $(2,3)$, we will find the tangent line at $x=2$. The derivative $y' = \frac{d}{dx}(x^{3/2} + 3x + 1)$ is: $$y' = \frac{3}{2}x^{1/2} + 3.$$ 5. Evaluate $y'(2)$: $$y'(2) = \frac{3}{2}(\sqrt{2}) + 3 = \frac{3}{2} \times 1.414 + 3 = 2.121 + 3 = 5.121.$$ This is the slope of the tangent line at $x=2$. 6. Compute the actual $y$ at $x=2$ from the function: $$y(2) = 2^{3/2} + 3(2) + 1 = 2.828 + 6 + 1 = 9.828.$$ So the point on the curve at $x=2$ is $(2, 9.828)$. 7. Equation of a line with slope $m$ through point $(x_0, y_0)$ is: $$y - y_0 = m(x - x_0).$$ Substituting $m=5.121$, $x_0=2$, and $y_0=9.828$ gives: $$y - 9.828 = 5.121(x - 2).$$ 8. Simplify the tangent line equation: $$y = 5.121x - 10.242 + 9.828 = 5.121x - 0.414.$$ **Final answer:** The equation of the tangent line to the curve $y = x^{3/2} + 3x + 1$ at $x=2$ is $$y = 5.121x - 0.414.$$