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**bakanagaijin** Consider the differential equation $\displaystyle \frac{dy}{dx} = 5x^2 - \frac{6}{y-2}$ for $\displaystyle y \neq 2$. Let $\displaystyle y=f(x)$ be the particular solution to this differential equation with the initial condition $\displaystyle f(-1) = -4$.

a. Evaluate $\displaystyle \frac{dy}{dx}$ and $\displaystyle \frac{d^2 y}{dx^2}$ at $\displaystyle (-1, -4)$

b. Is it possible for the x-axis to be tangent to the graph of $\displaystyle f$ at some point? Explain why or why not.

c. Find the second-degree Taylor polynomial for $\displaystyle f$ about $\displaystyle x=-1$.

d. Use Euler's method, starting at $\displaystyle x=-1$ with two steps of equal size, to approximate $\displaystyle f(0)$. Show the work that leads to your answer.

Thank you to anyone can help!