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3.4 Concavity and the Second Derivative Test

3.4 Concavity and the Second Derivative Test. Definition of Concavity. Let f be differentiable on an open interval I. The graph of f is concave upward on I if f’ is increasing on the interval and concave downward of I if f’ is decreasing on the interval. The graph is concave up

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3.4 Concavity and the Second Derivative Test

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  1. 3.4 Concavity and the Second Derivative Test

  2. Definition of Concavity • Let f be differentiable on an open interval I. The graph of f is concave upwardon I if f’ is increasing on the interval and concave downward of Iif f’ is decreasing on the interval. The graph is concave up if it all lies above the tangent lines The graph is concave down if it all lies below the tangent lines

  3. Test for concavity • Let f be a function whose second derivative exists on an open interval I. • 1. If f’’(x) > 0 for all x in I, then the graph of f is concave upward in I. • 2. If f’’(x) < 0 for all x in I, then the graph of f is concave downward in I. To apply the above theorem, locate x values at which f’’(x) = 0 or f’’(x) does not exist.

  4. Determine the intervals for concavity for the following f’’(x)=0 when x=1,-1 **NOTE** if there are any points where the original functions are discontinuous those values should be used for intervals as well.

  5. Determine the intervals for concavity for the following Notice that there are no values for x that will give f’’(x)=0, but there are points of discontinuity of 2,-2 for the original function. So use those points as your interval markers.

  6. Find the intervals on which b(x) = 3x3 + 2x2 -3 is concave up or down Up: (-∞,-2/9 ) and Down: (-2/9 ,∞)

  7. Find the intervals on which d(x) = x3 -6x2 + 9x -4 is concave up or down Up: (2,∞) and Down: (-∞,2)

  8. Points in which concavity changes are called inflection points. • Definition of Point of Inflection • Let f be a function that is continuous on an open interval and let c be a point in the interval. If the graph of f has a tangent line at this point (c,f(c)), then this point is a point of inflection of the graph f if the concavity of f changes from upward to downward (or downward to upward) at the point.

  9. Theorem for Inflection Points • If (c,f(c)) is a point of inflection of the graph of f, then either f’’(c)=0 or f’’ does not exist at x=c f’’(x)=0 at x=0 and x=2 so those are 2 possible inflection points, check the concavity to verify that the concavity changes from upward to downward (downward to upward) to see if they are inflection points.

  10. Find all inflection points of w(x) = x3 + 6x2 -4x + 1 x = -2

  11. Find all inflection points of c(x) = x4 + 6x3 -3x + 1 x = -3, x = 0

  12. “2nd Derivative Test” • Let f be a function such that f’(c) = 0 and the second derivative of f exists on an open interval containing c. • 1. If f’’(c) > 0, then f has a relative minimum at (c,f(c)). • 2. If f’’(c) < 0, then f has a relative maximum at (c,f(c)). If f’’(c) = 0, the test fails. That is, f may have a relative maximum, a relative minimum, or neither. In such cases, you can us the First Derivative Test.

  13. Apply the second derivative test on g(x) = -2x2 -3x to find all local maximum and minimums.

  14. Homework pg. 19511-15

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