1. We are asked to analyze and understand the function $$h(x) = \ln(x) + 6$$. 2. The function consists of the natural logarithm $$\ln(x)$$ which is defined for $$x > 0$$, shifted vertically upwards by 6 units. 3. To find the intercepts: - **x-intercept**: Solve $$h(x) = 0$$. $$\ln(x) + 6 = 0$$ $$\ln(x) = -6$$ $$x = e^{-6}$$ So, the x-intercept is at $$\left(e^{-6}, 0\right)$$. - **y-intercept**: Since $$\ln(x)$$ is undefined for $$x \leq 0$$, there is no y-intercept. 4. To find extrema, we differentiate: $$h'(x) = \frac{1}{x}$$. Since $$h'(x) > 0$$ for all $$x > 0$$, the function is strictly increasing and has no local maxima or minima. 5. Summary: $$h(x) = \ln(x) + 6$$ is an increasing logarithmic function with domain $$x > 0$$, x-intercept at $$\left(e^{-6}, 0\right)$$, no y-intercept, and no local extrema.