Subjects calculus

Local Extrema 4Bc5Ca

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1. **Problem Statement:** Find the absolute maximum and minimum values of the function $y=f(x)$ using the given local maxima and minima points. 2. **Given Information:** - Local maxima at $(2,6)$ and $(3,4)$ - Local minima at $(-2,1)$ and $(0,0)$ - The graph starts below zero and ends downward to the right. 3. **Step 1: Identify local maxima and minima values.** - Local maxima values: $f(2)=6$, $f(3)=4$ - Local minima values: $f(-2)=1$, $f(0)=0$ 4. **Step 2: Determine absolute maximum and minimum.** - Absolute maximum is the highest local maximum: $\max(6,4) = 6$ at $x=2$ - Absolute minimum is the lowest local minimum: $\min(1,0) = 0$ at $x=0$ 5. **Step 3: Check if any other absolute extrema exist.** - The function starts below zero but no exact value given, so cannot confirm absolute minimum lower than 0. - The function ends downward but no higher value than 6 found. 6. **Conclusion:** - Absolute maximum value is $f(2)=6$ - Absolute minimum value is $f(0)=0$ - Local minima are at $(-2,1)$ and $(0,0)$ - Local maxima are at $(2,6)$ and $(3,4)$ **Answer:** Option B is correct: There are two local minima. The leftmost minimum is $f(-2)=1$ and the rightmost minimum is $f(0)=0$.