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) = 3x^4 + 16x^3 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 numbers in the expression. Use a comma to separate answers as needed.) There are no relative extrema.

Explanation / Answer

points of inflection => d^2y/dx^2 =0

dy/dx = 12x^3+48x^2

d^2y/dx^2 = 36 x^2 +96 x = 0

=> x= 0 and x= -96/36 =-8/3 , so x=0 and x=-8/3 are points of inflextion ( put x in y to get y coordinate)

for coordinates of extrema we have

dy/dx =0 => 12x^3+48x^2 =0

=> x^2( 12+48x) =0

=> x= 0 and x = -4

corresponding y coordinates are (3*0+16*0) and (3*(-4)^4+16*(-4)^3)

so extremum points are (0,0) and (-4,-256) but at (0,0) we have inflextion so

only (-4,-256) is the extremum point

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