Sketch the graph of the following function. List the coordinates of where extrem

Question

Sketch the graph of the following function. List the coordinates of where extrema or points of inflection occur. State where the function in increasing or decreasing as well as where it is concave up or concave down. f(x) = x^4 - 4x^3 + 2 What are the coordinates of the relative extrema? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The coordinates of the relative extrema are. (Simplify your answer. Type an ordered pair. Type an exact answer using radicals as needed. Use integers or fractions for any number in the expression. Use a comma to separate answers as needed.) There are no relative extrema.

Explanation / Answer

f(x) = x4 - 4x3 + 2

Critical points ==> f '(x) =0

==> 4x4-1 - 4(3)x3-1 + 0 = 0

==> 4x3 - 12x2 = 0

==> 4x2(x - 3) = 0

==> x = 0 , x = 3

f ''(x) = 4(3)x3-1 - 12(2)x2-1 = 12x2 - 24x

f ''(0) = 12(0)2 - 24(0) = 0

As f ''(x) = 0 at x = 0 ==> at x = 0 the function has inflection point

x = 0 ==> y = 2 . Hence inflection point is (0 , 2)

f ''(3) = 12(3)2 - 24(3) = 36 > 0

As f ''(x) > 0 , function has local minimum at x = 3

x = 3 ==> y = 34 - 4(3)3 + 2 = -25

hence local minima at (3 , -25)

Local maxima does not exist

f''(x) = 0

==> 12x2 - 24x = 0

==> 12x(x -2 ) = 0

==> x = 2

x = 2 ==> f(x) = 24 - 4(2)3 + 2 = -14

Hence (2 , -14) is an inflection point

Hence local minima at (3 , -25) and inflection points at (0 , 2) and (2, -14)

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