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# Sage Application Assignment: Derivatives and Graphing
# Stephanie Munera
# December 10, 2018

Given f(x)=(1/6)x4−2x2+4f(x)= (1/6)x^4-2x^2+4:

  1. Determine the first derivative of the function.

f(x)=(1/6)*x^4-2*x^2+4

# f.derivative()
# show(f.derivative())

# diff(f(x), x)
# show(diff(f(x)))

  1. Use the first derivative to determine the local maximum and local minimum of the function. How can you use these points to determine where the function is increasing or decreasing?

# plot((2/3)*x^3-4*x, (x, -5, 5), ymax = 5, ymin = -5, color='red') + points([(0,0), (-sqrt(6), 0), (sqrt(6), 0)], color='red', pointsize=25)

  1. Determine the second derivative of the function.

g(x)=(2/3)*x^3-4*x

# g.derivative()
# show(g.derivative())

f(x)=(1/6)*x^4-2*x^2+4

# diff(f(x), x, 2)
# show(diff(f(x), x, 2))

  1. Use the second derivative of the function to determine the inflection points of the function. How can you use these points to determine where the function is concave up or concave down?

# plot(2*x^2-4, (x, -5, 5), ymax = 5, ymin = -5, color='green') + points([(-sqrt(2), 0), (sqrt(2), 0)], color='green', pointsize=25)

  1. Using the information from the problems above, graph the function.

# plot((1/6)*x^4-2*x^2+4, (x, -5, 5), ymax = 5, ymin = -5, color='blue') + points([(0,4)], color='blue', pointsize=25) + points([(-sqrt(2), (2/3)), (sqrt(2), (2/3))], color='blue', pointsize=25) + points([(-sqrt(6), -2), (sqrt(6), -2)], color='blue', pointsize=25)

# local maximum
# points([(0,4)], color='blue', pointsize=25)

# local minimum
# points([(-sqrt(6), -2), (sqrt(6), -2)], color='blue', pointsize=25)

# inflection points
# points([(-sqrt(2), (2/3)), (sqrt(2), (2/3))], color='blue', pointsize=25)

# plot((2/3)*x^3-4*x, (x, -5, 5), ymax = 5, ymin = -5, color='red') + points([(0,0), (-sqrt(6), 0), (sqrt(6), 0)], color='red', pointsize=25)
# plot(2*x^2-4, (x, -5, 5), ymax = 5, ymin = -5, color='green') + points([(-sqrt(2), 0), (sqrt(2), 0)], color='green', pointsize=25)