Answer :

We find critical points at x = -1 and x = 0 by analyzing the derivative f'(x). By evaluating f' at intervals, we see a relative maximum at x = -1 and a relative minimum at x = 0. Due to the high power of x, the function approaches positive infinity as x goes to positive or negative infinity, so it has no absolute extrema.

We can find the extrema (i.e. maximum and minimum points) of the function by first finding its critical points and then analyzing the function's behavior around those points.

Steps to solve:

1. Find the derivative of the function:

f'(x) = 63x^{8} - 63x^{6}

2. Find the critical points:

The critical points are the values of x where the derivative is zero or undefined. In this case, the derivative is a polynomial and is defined for all real numbers. Therefore, the critical points occur where f'(x) = 0:

63x^{8} - 63x^{6} = 0

Factoring out 63x^{6}, we get:

63x^{6} (x^2 - 1) = 0

This gives us two critical points: x = 0 and x = ±1.

3. Analyze the intervals:

Since the derivative is a polynomial, it's defined for all real numbers. Therefore, we can analyze our function's behavior by looking at the intervals defined by the critical points:

- Interval 1: x < -1

- Interval 2: -1 < x < 0

- Interval 3: x > 0

4. Evaluate the derivative at each interval:

Let's evaluate f'(x) at each interval to see if it's positive or negative on that interval. A positive derivative indicates an increasing function, and a negative derivative indicates a decreasing function.

- Interval 1 (x < -1): f'(-2) = 63(-2)^{8} - 63(-2)^{6} > 0 (positive)

- Interval 2 (-1 < x < 0): f'(-0.5) = 63(-0.5)^{8} - 63(-0.5)^{6} < 0 (negative)

- Interval 3 (x > 0): f'(1) = 63(1)^{8} - 63(1)^{6} > 0 (positive)

5. Determine extrema:

Now, let's look at the critical points:

- x = -1: Since the function changes from increasing to decreasing at x = -1, it has a relative maximum at this point.

- x = 0: Since the function changes from decreasing to increasing at x = 0, it has a relative minimum at this point.

6. Absolute extrema:

We also need to consider the function's behavior at the edges of the defined domain (which is all real numbers in this case).

- As x approaches positive or negative infinity, f(x) approaches positive infinity due to the high power of x in the terms. Therefore, the function doesn't have an absolute maximum or minimum.

The function f(x) = 7x⁹ - 9x⁷ - 7 has:

- A relative maximum at x = -1.

- A relative minimum at x = 0.

- No absolute maximum or minimum.

Thanks for taking the time to read Find the absolute maximum and absolute minimum values of the function f x 7x 9 9x 7 7. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada