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  1. You know that the first derivative, f ′(x), tells us the slope and the rate of change of f(x). Homework Help ✎

    1. What does the second derivative, f ″(x), tell you about f ′(x)? What does f ″(x) tell you about f(x)?

    2. Find f ′(x) and f ″(x) for f(x) = x3 + 3x2 − 9x + 2.

    3. Use these derivatives to find f ″( l ) and f ″(−2). Is f(x) getting steeper or less steep at x = 1? At x = −2? Explain your reasoning.

    4. The values in part (c) can be used to determine concavity. Where is f(x) concave up? Where is f(x) concave down?

f ''(x) tells us the same thing about f '(x) that f '(x) tells us about f(x).

f '(x): 3x² + 6x −9
f ''(x): 6x + 6

Consider this: Even though a slope of −2 is steeper than a slope of −1, if the slope changes from −1 to −2, then the slope is decreasing.

Positive values of the 2nd-derivative indicate that the function is concave up while negative values indicate that the function is concave down.