微分應用+畫圖,,f4,,

已知曲線C1:y=x(2x+3)^1/2

a.試求使曲線有定義的x範圍。
2x + 3 => 0
x => -3/2

b.試求曲線的x軸截距及y軸截距。
when y = 0,
x(2x + 3)^1/2 = 0
x = 0 or 2x + 3 = 0
x = 0 or x = -3/2
when x = 0,
y = 0
so,C1 passes through (0,0) , (-3/2,0)

c.試求其導數及二階導數。並由此求轉向點,並指出各點是極大點還是極小點。
y = x(2x + 3)^0.5
dy/dx = x[0.5[(2x + 3)^-0.5] (2)] + (2x + 3)^0.5
dy/dx = 3(x + 1) / [(2x + 3)^0.5]
d^2y/dx^2 = 3(x + 2)/[(2x + 3)^1.5]
when dy/dx = 0,
x = -1,y = -1
d^2y/dx^2|[x = -1] > 0,
so,(-1,-1) is minimum.
there is no maximum since when x -----> infinity,y---->infinity
also,there is not the given range

d.描繪曲線c1的圖像。
C1 starts from (-3/2,0)
then it decreases to (-1,-1)
then it increases to (0,0)
after that,C1 is increasing.

e.由此,描繪C2:y=|x|(2x+3)^1/2
just reflect C1 which the part under the x-axis to above the x-axis.