Determine the x-coordinate(s)of any relative extrema and inflection points of the function .
A)relative minimum: ;inflection point:
B)no relative extrema;inflection point:
C)relative minimum: ;no inflection points
D)no relative extrema;inflection point:
E)relative maximum: inflection point:

Question

Determine the x-coordinate(s)of any relative extrema and inflection points of the function  . ​
A)relative minimum: ;inflection point: 
B)no relative extrema;inflection point: 
C)relative minimum: ;no inflection points 
D)no relative extrema;inflection point: 
E)relative maximum: inflection point:

Quizplus · Accepted Answer

To find relative extrema and inflection points, we need to take the first and second derivative of the given function. Taking the derivative of the given function 11eac44f_ed48_4c07_bedf_99f196d96e00_TB7497_11, we get:

f(x) = 11eac44f_ed48_4c07_bedf_99f196d96e00_TB7497_11
f'(x) = 44eac44f_ed48_4c09_bedf_d2522a25a528_TB7497_11
f''(x) = 198eac44f_ed48_4c0a_bedf_723c987ed9f9_TB7497_11

Setting f'(x) = 0, we get x = 0. This means that there is a critical point at x = 0. To check whether it is a relative maximum or minimum, we need to check the sign of f''(x) at x = 0. Since f''(0) = 198eac44f_ed48_4c0a_bedf_723c987ed9f9_TB7497_11 is positive, we can conclude that x = 0 is a relative minimum.

Next, we need to find the inflection points. For this, we need to set f''(x) = 0 and solve for x.  Setting f''(x) = 0, we get x = 0 again. However, this is not an inflection point because its sign does not change on either side of x = 0. Therefore, there are no inflection points.

Hence, the best choice is C) relative minimum: 11eac44f_ed48_9a2b_bedf_131bf315224a_TB7497_11 ;no inflection points.

Determine the X-Coordinate(s)of Any Relative Extrema and Inflection Points of the Function