How do you sketch the curve #y=x^2+1/x# by finding local maximum, minimum, inflection points, asymptotes, and intercepts?

Question

William Abdalla · Accepted Answer

See the answer below:

Evelyn Agard · Answer

undefined To sketch the curve $y = x^2 + \frac{1}{x}$ and find its local maximum, minimum, inflection points, asymptotes, and intercepts:

1. **Intercepts**:
   - **x-intercept**: Set $y = 0$ and solve for $x$.
   - **y-intercept**: Set $x = 0$ and solve for $y$.

2. **Asymptotes**:
   - **Horizontal asymptote**: As $x$ approaches positive or negative infinity, $y$ approaches the horizontal asymptote, which is determined by the highest power term in the expression.
   - **Vertical asymptotes**: Set the denominator of the rational expression equal to zero and solve for $x$.

3. **Local Maximum and Minimum**:
   - Take the derivative of $y$ with respect to $x$ and find critical points by setting the derivative equal to zero and solving for $x$.
   - Use the second derivative test to classify the critical points as local maxima, minima, or points of inflection.

4. **Inflection Points**:
   - Find the second derivative of $y$ with respect to $x$.
   - Set the second derivative equal to zero and solve for $x$ to find points of inflection.

5. **Sketch the Curve**:
   - Use the information obtained above to sketch the curve, including intercepts, asymptotes, and critical points.

Follow these steps systematically to sketch the curve and locate the desired points.