a graphing calculator is recommended.\n(a) the curve with equation ( y^{2}=x^{3}+3 x^{2} ) is called the…

a graphing calculator is recommended.\n(a) the curve with equation ( y^{2}=x^{3}+3 x^{2} ) is called the tschirnhausen cubic. find an equation of the tangent line to this curve at the point ( (1,2) ).\n( y=)\n(b) at what points does this curve have horizontal tangents? (order your answers from smallest to largest ( x ), then from smallest to largest ( y ).)\n( (x, y)=(quad) )\n( (x, y)=(quad) )\n(c) illustrate parts (a) and (b) by graphing the curve and the tangent lines on a common screen.

a graphing calculator is recommended.\n(a) the curve with equation ( y^{2}=x^{3}+3 x^{2} ) is called the tschirnhausen cubic. find an equation of the tangent line to this curve at the point ( (1,2) ).\n( y=)\n(b) at what points does this curve have horizontal tangents? (order your answers from smallest to largest ( x ), then from smallest to largest ( y ).)\n( (x, y)=(quad) )\n( (x, y)=(quad) )\n(c) illustrate parts (a) and (b) by graphing the curve and the tangent lines on a common screen.

Answer

Explanation:

Step1: Differentiate the equation implicitly

Differentiate (y^{2}=x^{3}+3x^{2}) with respect to (x). Using the chain - rule ((y^{2})^\prime = 2y\frac{dy}{dx}) and the power - rule ((x^{n})^\prime=nx^{n - 1}), we get: (2y\frac{dy}{dx}=3x^{2}+6x). Then (\frac{dy}{dx}=\frac{3x^{2}+6x}{2y}).

Step2: Find the slope of the tangent line at the point ((1,2))

Substitute (x = 1) and (y = 2) into (\frac{dy}{dx}). (\frac{dy}{dx}\mid_{(x = 1,y = 2)}=\frac{3(1)^{2}+6(1)}{2(2)}=\frac{3 + 6}{4}=\frac{9}{4}).

Step3: Use the point - slope form of a line (y - y_{1}=m(x - x_{1}))

Here (x_{1}=1,y_{1}=2,m=\frac{9}{4}). (y - 2=\frac{9}{4}(x - 1)). Expand: (y-2=\frac{9}{4}x-\frac{9}{4}). (y=\frac{9}{4}x-\frac{9}{4}+2=\frac{9}{4}x-\frac{9}{4}+\frac{8}{4}=\frac{9}{4}x-\frac{1}{4}).

Step4: Find the points with horizontal tangents

A horizontal tangent occurs when (\frac{dy}{dx}=0). Since (\frac{dy}{dx}=\frac{3x^{2}+6x}{2y}), set (3x^{2}+6x = 0). Factor out (3x): (3x(x + 2)=0). So (x=0) or (x=-2). When (x = 0), then (y^{2}=0^{3}+3(0)^{2}=0), so (y = 0). When (x=-2), then (y^{2}=(-2)^{3}+3(-2)^{2}=-8 + 12 = 4), so (y=\pm2).

Answer:

(a) (y=\frac{9}{4}x-\frac{1}{4}) (b) ((x,y)=(-2,-2)), ((x,y)=(-2,2)), ((x,y)=(0,0))