1. **Problem Statement:** We are given a continuous function $y = x^2$ and a graph with points labeled A, B, C, D, E, G, H, J showing local maxima and minima. We need to explain local extrema, how they differ from absolute maxima and minima, and determine intervals of increase and decrease around these points. 2. **Definitions and Formulas:** - A **local maximum** at $x = c$ means $f(c)$ is greater than values of $f(x)$ near $c$. - A **local minimum** at $x = c$ means $f(c)$ is less than values of $f(x)$ near $c$. - An **absolute maximum** is the highest value of $f(x)$ on the entire domain. - An **absolute minimum** is the lowest value of $f(x)$ on the entire domain. 3. **Explanation:** - Local extrema are points where the function changes direction from increasing to decreasing (max) or decreasing to increasing (min). - Absolute extrema are the highest or lowest points over the entire domain, not just nearby points. 4. **Given Graph Points and Extrema:** - Local maxima near points A, C, E, G, J. - Local minima near points B, D, H. 5. **Intervals of Increase and Decrease:** - Between points A and B: function decreases (from max at A to min at B). - Between points B and C: function increases (from min at B to max at C). - Between points C and D: function decreases (max at C to min at D). - Between points D and E: function increases (min at D to max at E). - Between points E and G: function decreases (max at E to max at G, but since both are maxima, function decreases then increases, so more detail needed). - Between points G and H: function decreases (max at G to min at H). - Between points H and J: function increases (min at H to max at J). 6. **Summary of intervals:** - $(A, B)$: decreasing - $(B, C)$: increasing - $(C, D)$: decreasing - $(D, E)$: increasing - $(E, G)$: oscillating, but generally decreasing then increasing - $(G, H)$: decreasing - $(H, J)$: increasing 7. **Relation to $y = x^2$ and $ heta = (0,0)$:** - The function $y = x^2$ has a single minimum at $(0,0)$, which is an absolute minimum. - The graph described oscillates, so it is not $y = x^2$ but a different oscillating function. **Final note:** Local extrema are points where the function changes direction locally, while absolute extrema are the highest or lowest values over the entire domain.