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Here is the quick direct breakdown:
1 Identify cr
Critical points occur where f'(x) = 0:
(a) Critical points: A, C, E, G
2 Determine intervals of increase and decrease
f(x) increases where f'(x) > 0 (above the x-axis) and decreases where f'(x) < 0 (below the x-axis):
(b) Increasing: (A, C) U (E, G)
Decreasing: (-∞, A) U (C, E) U (G, ∞)
3 Find local extrema
(c) Local minima (f' changes from negative to positive): A, E
(d) Local maxima (f' changes from positive to negative): C, G
4 Determine concavity and inflection points
Concave up is where f'(x) is increasing; concave down is where f'(x) is decreasing:
(e) Concave Up: (-∞, B) U (D, F)
**Concav
(f) Inflection points (turning points of f'): B, D, F
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Explanation
1 Identify critical points and sign of f'(x)
f'(x) = 0 => x ∈ {A, C, E, G}
f'(x) > 0 => x ∈ (A, C) U (E, G)
f'(x) < 0 => x ∈ (-∞, A) U (C, E) U (G, ∞)
2 Determine local extrema from sign
Critical Points and Sign of f'(x)
The derivative crosses the horizontal axis at A, where f prime equals zero.
It also equals zero at point C.
Another zero occurs at point E, giving our critical points.