The image you've uploaded presents a question where you are tasked with:

1. Verifying whether the point (7, 7) lies on the curve defined by the equation $x^3 + y^3 = 14xy$.
2. Determining the equation of the tangent line to the curve at the point (7, 7).

Part a: Verify that the point (7, 7) lies on the curve

To verify if the point $(7, 7)$ lies on the curve, substitute $x = 7$ and $y = 7$ into the given equation $x^3 + y^3 = 14xy$.

$7^3 + 7^3 = 14(7)(7)$ $343 + 343 = 14 \times 7 \times 7$ $686 = 686$

Since both sides are equal, the point $(7, 7)$ lies on the curve.

Part b: Find the equation of the tangent line at (7, 7)

To find the tangent line, we first need to differentiate the given equation implicitly to find $\frac{dy}{dx}$. Differentiating both sides of the equation $x^3 + y^3 = 14xy$ with respect to $x$:

$\frac{d}{dx}(x^3) + \frac{d}{dx}(y^3) = \frac{d}{dx}(14xy)$ $3x^2 + 3y^2 \frac{dy}{dx} = 14y + 14x \frac{dy}{dx}$

Now, we solve for $\frac{dy}{dx}$ by isolating it:

$3y^2 \frac{dy}{dx} - 14x \frac{dy}{dx} = 14y - 3x^2$ $\left(3y^2 - 14x\right) \frac{dy}{dx} = 14y - 3x^2$ $\frac{dy}{dx} = \frac{14y - 3x^2}{3y^2 - 14x}$

Now, substitute $x = 7$ and $y = 7$ into this equation to find the slope of the tangent line at $(7, 7)$:

$\frac{dy}{dx} = \frac{14(7) - 3(7^2)}{3(7^2) - 14(7)}$ $\frac{dy}{dx} = \frac{98 - 147}{147 - 98}$ $\frac{dy}{dx} = \frac{-49}{49} = -1$

So, the slope of the tangent line at $(7, 7)$ is $-1$.

Finally, the equation of the tangent line can be written using the point-slope form:

$y - y_1 = m(x - x_1)$ Substitute $m = -1$, $x_1 = 7$, and $y_1 = 7$:

$y - 7 = -1(x - 7)$ $y - 7 = -x + 7$ $y = -x + 14$

So, the equation of the tangent line at $(7, 7)$ is:

$y = -x + 14$

Let me know if you'd like further explanation or have any questions! Here are some related questions:

1. How would the tangent line change if the point was at $(3, 5)$ instead?
2. How do implicit differentiation and explicit differentiation differ when solving such problems?
3. What are other methods to find tangent lines besides implicit differentiation?
4. How does the graph of the equation $x^3 + y^3 = 14xy$ behave in general?
5. What does the result of $\frac{dy}{dx} = -1$ tell us about the slope of the curve?

Tip: When differentiating implicitly, remember to apply the chain rule carefully, especially when dealing with terms involving $y$.