Extrema occur at critical points; critical points are either where the first derivative fails to exist or equals 0.
Horizontal tangent lines occur where the first derivative is 0.
Points of inflections occur when the concavity changes across a point.
The Attempt at a Solution
14) Yes, the original function has a horizontal tangent line at x = 2. The graph of the derivative is 0 at the specified point. The original function also has an inflection point because the derivative of the derivative changes sign from positive to negative.
15) No, the derivative is negative at x = -2 (the second tick mark on the x-axis to the left of the origin). It is concave up however since the derivative of the derivative is positive.
16) x = 0 is a critical point since the derivative is going to the infinities. However, the derivative to the left is positive; the derivative to the left is negative. It's a local maximum.
17) Yes; the second derivative is negative only from 2 to infinity.
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