▶️ Answer/Explanation
Solution
(a) Absolute Extrema:
- Absolute maximum at \( x = 0 \) because:
- \( f(0) = 1 \) is the highest value in the interval
- Function decreases on both sides (since \( f”(x) < 0 \) implies concave down)
- Absolute minima at \( x = \pm 2 \) because:
- \( f(\pm 2) = -1 \) is the lowest value
- Function changes from decreasing to increasing at \( x = 2 \) (as \( f” \) changes from negative to positive)
- By even symmetry, same at \( x = -2 \)
(b) Points of Inflection:
- At \( x = \pm 1 \) because:
- \( f”(1) = 0 \) and changes concavity pattern
- For \( 0 < x < 1 \): \( f”(x) < 0 \) (concave down)
- For \( 1 < x < 2 \): \( f”(x) < 0 \) but \( f”'(x) \) changes from positive to negative
- The third derivative change indicates the concavity reaches an extremum
(c) Graph Characteristics:
- Even symmetry about y-axis
- Peak at (0,1) with smooth curve
- Passes through (1,0) with inflection
- Minimum at (2,-1) with vertical tangent (undefined \( f” \))
- Concave up for \( x > 2 \)
