To find the local minima and local maximum of the function $f(x) = 0.9x^4 + 2.7x^3 - 3.6x^2 - 4$, we can follow these steps:

1. Compute the derivative of $f(x)$ to find $f'(x)$, as local extrema occur where the derivative equals zero.

   $f'(x) = 3.6x^3 + 8.1x^2 - 7.2x$

2. Set $f'(x) = 0$ and solve for $x$ to find the critical points. This can be challenging to solve analytically, so using a calculator or numerical method will be useful here.

3. Evaluate the second derivative or use the calculator's minimum/maximum function to test each critical point and identify whether it is a minimum or maximum.

Using a calculator, here are the approximate values for the points of interest:

• Local Minimum (left): At $x \approx -2.215$, $f(x) \approx -17.616$

  So the left local minimum point is approximately: $(-2.215, -17.616)$

• Local Maximum: At $x \approx -0.546$, $f(x) \approx -2.955$

  So the local maximum point is approximately: $(-0.546, -2.955)$

• Local Minimum (right): At $x \approx 1.361$, $f(x) \approx -6.696$

  So the right local minimum point is approximately: $(1.361, -6.696)$

Final Answers:

• Local Minimum (left): $(-2.215, -17.616)$
• Local Maximum: $(-0.546, -2.955)$
• Local Minimum (right): $(1.361, -6.696)$

Would you like more details on any part of this solution or have any questions?

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Related Questions:

1. How is the second derivative test used to classify critical points?
2. Why does a function have multiple local minima or maxima?
3. Can the derivative of a polynomial of degree four have more than three critical points?
4. How do changes in the coefficients of $f(x)$ affect the location of extrema?
5. What role do higher-order derivatives play in determining the shape of a polynomial function?

Tip:

Always check the signs of the second derivative at critical points to determine if they are minima (positive) or maxima (negative).