First and Second Derivatives (1) - 12.1\/12.3 First Derivatives and Second Derivatives Consider a function and a point on the graph(a f(a a b c d If

First and Second Derivatives (1) - 12.1/12.3 First...

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12.1/12.3 First Derivatives and Second Derivatives Consider a function, 𝑓(𝑥) and a point on the graph, (a, f(a)). a) If 𝑓 (𝑎) > 0 then the function is increasing through the point. b) If 𝑓 (𝑎) < 0 then the function is decreasing through the point. c) If 𝑓 (𝑎) = 0 then the function has a stationary point at 𝑥 = 𝑎. d) If 𝑓 (𝑎) is undefined then the function has a singular point at 𝑥 = 𝑎. Stationary Points and Singular Points are collectively known as Critical Points. First Derivative Test for Extreme Points Consider that 𝑓(𝑥) has a Critical Point at 𝑥 = 𝑎 : 1) If 𝑓 (𝑥) > 0 at points to the left of the Critical Point and 𝑓 (𝑥) < 0 at points to the right of the Critical Point, then f(x) has a Relative Maximum at 𝑥 = 𝑎. 2) If 𝑓 (𝑥) < 0 at points to the left of the Critical Point and 𝑓 (𝑥) > 0 at points to the right of the Critical Point, then f(x) has a Relative Minimum at 𝑥 = 𝑎. 3) If the derivative is the same sign on either side of the Critical Point, then

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