Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$Harold tried to find all the points on the curve given by xy^(2)-x^(2)y=-54 where the line tangent to the curve is horizontal. This is his solution: Step 1: Finding an expression for (dy)/(dx). (dy)/(dx)=(y(2x-y))/(x(2y-x)) Step 2: Forming a system of equations. {[xy^(2)-x^(2)y=-54],[x(2y-x)=0],[y(2x-y)!=0]:} Step 3: Solving the system. (6,3) Is Harold's solution correct? If not, at which step did he make a mistake? Choose 1 answer: (A) The solution is correct. (B) Step 1 is incorrect. (C) Step 2 is incorrect. (D) Step 3 is incorrect.$

Harold tried to find all the points on the curve given by $x y^{2}-x^{2} y=-54$ where the line tangent to the curve is horizontal. This is his solution: $\newline$ Step $1$ : Finding an expression for $\frac{d y}{d x}$ . $\newline$ $\frac{d y}{d x}=\frac{y(2 x-y)}{x(2 y-x)}$ $\newline$ Step $2$ : Forming a system of equations. $\newline$ $\left\{\begin{array}{l} x y^{2}-x^{2} y=-54 \\ x(2 y-x)=0 \\ y(2 x-y) \neq 0 \end{array}\right.$ $\newline$ Step $3$ : Solving the system. $\newline$ $(6,3)$ $\newline$ Is Harold's solution correct? If not, at which step did he make a mistake? $\newline$ Choose $1$ answer: $\newline$ (A) The solution is correct. $\newline$ (B) Step $1$ is incorrect. $\newline$ (C) Step $2$ is incorrect. $\newline$ (D) Step $3$ is incorrect.

View step-by-step help

Home

Math Problems

Algebra 1

Compare linear and exponential growth

Full solution

Q. Harold tried to find all the points on the curve given by

x y^{2}-x^{2} y=-54

where the line tangent to the curve is horizontal. This is his solution:

\newline

Step

1

: Finding an expression for

\frac{d y}{d x}

\newline

\frac{d y}{d x}=\frac{y(2 x-y)}{x(2 y-x)}

\newline

Step

2

: Forming a system of equations.

\newline

\left\{\begin{array}{l} x y^{2}-x^{2} y=-54 \\ x(2 y-x)=0 \\ y(2 x-y) \neq 0 \end{array}\right.

\newline

Step

3

: Solving the system.

\newline

(6,3)

\newline

Is Harold's solution correct? If not, at which step did he make a mistake?

\newline

Choose

1

answer:

\newline

(A) The solution is correct.

\newline

(B) Step

1

is incorrect.

\newline

2

is incorrect.

\newline

(D) Step

3

is incorrect.

Finding Expression for $(dy)/(dx)$ : Finding an expression for $(dy)/(dx)$ . To find where the tangent line to the curve is horizontal, we need to find where the derivative $(dy)/(dx)$ is equal to zero. Harold's expression for $(dy)/(dx)$ is: $(dy)/(dx) = \frac{y(2x-y)}{x(2y-x)}$ We need to verify this by differentiating the given equation $xy^2 - x^2y = -54$ implicitly with respect to $x$ . Differentiating both sides with respect to $x$ , we get: $y^2 + 2xy\frac{dy}{dx} - 2xy\frac{dy}{dx} - x^2\frac{dy}{dx} = 0$ Simplifying, we get: $y^2 - x^2\frac{dy}{dx} = 0$ Solving for $(dy/dx)$ , we get: $(dy/dx) = \frac{y^2}{x^2}$ It seems that Harold's expression for $(dy)/(dx)$ is incorrect.