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Finding Increasing & Decreasing Intervals When f’(c) = 0 or is undefined → c is a critical number Steps * Find the 1st derivative and set it equal to 0 * Solve for x to find the critical numbers * Use test points (tp) to determine the sign of f’(x) Example: f(x) = x2 If f’(x) < 0 then function is decreasing 2 X -9 on the interval (a,b) If f’(x) > 0 then function is increasing f’(x) = -18x = 0 on the interval (a,b) 2 2 If f’(x) = 0 then function is constant (x - 9) on the interval (a,b) -18x = 0 (x2 – 9)2 = 0 x = 0 → critical number x = -3, 3 → discontinuities -3 3 0 -10 -1 1 10 tp tp tp tp Use test points to find increasing and decreasing intervals f’(-10) = + → increasing f’(1) = − → decreasing f’(-1) = + → increasing f’(10) = − → decreasing Therefore: Increasing on (-∞, -3), (-3, 0) Decreasing on (0, 3), (3, ∞) Relative max: (0,0) Maximums and Minimums Max @ (a, f(a)) if: then x = a is a relative maximum (+) a (-) Min @ (b, f(b)) if: (+) (-) b (-) (+) then x = b is a relative minimum