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Solutions to Test 5
  1. If tex2html_wrap_inline71 Then
    1. Find the interval(s) where f is increasing and decreasing. Answer: Since tex2html_wrap_inline75 the critical numbers are at tex2html_wrap_inline77 and x=2. Note that f is not defined for x<2, and tex2html_wrap_inline85 in x>2. So f is decreasing in tex2html_wrap_inline91
    2. Find the maximum or minimum if any. Answer: The maximum will happen at the left end point, x=2, y=-4.
    3. Find tex2html_wrap_inline97 (Answer: tex2html_wrap_inline99).
    4. Sketch the graph of f. (Use the following Maple command to check your answer plot(-3*x*sqrt(x-2)-4,x=0..3,y=-10..0); ).
  2. If tex2html_wrap_inline103 then
    1. Find the interval(s) where f is increasing or decreasing. (Answer: tex2html_wrap_inline107 tex2html_wrap_inline109 in tex2html_wrap_inline111 and tex2html_wrap_inline85 in tex2html_wrap_inline115 So , f is increasing in tex2html_wrap_inline111 and f is decreasing in (-3, tex2html_wrap_inline125
    2. Find the maximum or minimum, if any. (Answer: f has a maximum at (-3,3).
    3. Find tex2html_wrap_inline131 (Answer: tex2html_wrap_inline133 )
    4. Find tex2html_wrap_inline135 (Answer: tex2html_wrap_inline137. )
    5. Find the interval(s) where f is concave upward or downward. (Answer, tex2html_wrap_inline141 tex2html_wrap_inline143 is always positive, so f is concave upward in tex2html_wrap_inline147
    6. sketch the graph of f. (Answer, use plot(-2*(x+3)^(2/3) +3, x=-4..4, y=-5..5);  note that Maple only give you half of the graph, you need to fix this).
  3. If f(x)=-3x5+5x3, then
    1. Find the interval(s) where f is increasing or decreasing. (Answer: tex2html_wrap_inline155 tex2html_wrap_inline85 in tex2html_wrap_inline159 so f is decreasing in tex2html_wrap_inline163 and tex2html_wrap_inline109 in (-1,1), so f is increasing in (-1,1).
    2. Find the maximum or minimum, if any. (Answer: f has a minimum at x=-1, y=-2, and maximimum at x=1, y=2).
    3. Assume tex2html_wrap_inline143 can be factored into tex2html_wrap_inline185 then find the interval(s) where f is concave upward and concave downward. (Answer, f is concave upward in tex2html_wrap_inline191 and concave downward in tex2html_wrap_inline193
    4. Find the inflection point of f. (Answer, the inflection points of f are at tex2html_wrap_inline199 and tex2html_wrap_inline201
    5. Sketch the graph of f. (Use plot(-3*x^5+5*x^3,x=-2..2, y=-5..5); to verify your answer)
  4. If f(x)=x3-6x2+2, the
    1. Find the interval(s) where f is increasing or decreasing. (Answer: tex2html_wrap_inline209 tex2html_wrap_inline109 in tex2html_wrap_inline213 so f is increasing in tex2html_wrap_inline213 and tex2html_wrap_inline85 in (0,4) so f is decreasing in (0,4).)
    2. Find the maximum or minimum, if any. (Answer: f has a maximum at x=0,y=2 and a minimum at x=4,y=-30).
    3. Find the interval(s) where f is concave upward and concave downward. (Answer: tex2html_wrap_inline235 f is cocave upward in tex2html_wrap_inline239 and f is concave downward in tex2html_wrap_inline243
    4. Find the inflection point of f. (Answer: the inflection point of f is at (2,-14).
    5. Sketch the graph of f. (Use plot(x^3-6*x^2+2, x=-2..7, y=-40..10); to verify your answer).


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