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Examples

Example: For the function f(x)=3x5-5x3. Sketch the graph by finding the followings: the intervals that f is increasing or decreasing. Since we have tex2html_wrap_inline104 and we observe the following tex2html_wrap_inline106 We see that f is increasing in tex2html_wrap_inline110 and decreasing in tex2html_wrap_inline112

relative extrema: f has a relative maximum at x=-1, and f has a relative minimum at x=1. The interval that f is concave up or concave down: We observe tex2html_wrap_inline124tex2html_wrap_inline126Therefore the function f is concave upward on tex2html_wrap_inline130 and concave downward on tex2html_wrap_inline132 The inflection points of f are at tex2html_wrap_inline136 and tex2html_wrap_inline138 For graph, use plot(3*x^5-5*x^3,x=-2..2,y=-5..5, thickness=3); Pay attention the intervals where f is concave upward or downward.