Solving the Equation (a^5 - a^3a 2) and Determining the Range of (a^6)

Suppose (a) is a positive real number such that (a^5 - a^3a 2). What is the range of (a^6)?

Step-by-Step Solution

To solve the given equation (a^5 - a^3a 2), we can start by rewriting it as:

a^5 - a^3a - 2 0.

We define a function (f(a) a^5 - a^3 - 2).

Step 1: Analyzing the Function (f(a))

Let's evaluate (f(a)) at a few key points:

When (a 1):

f(1) 1^5 - 1^3*1 - 2 1 - 1*1 - 2 -1.

When (a 2):

f(2) 2^5 - 2^3*2 - 2 32 - 8*2 - 2 24.

Since (f(1) 0), by the Intermediate Value Theorem, there is at least one root in the interval ([1, 2]).

Step 2: Finding the Derivative (f'(a))

To understand the behavior of (f(a)), we compute its derivative:

f'(a) 5a^4 - 3a^2.

Now, we analyze (f'(a)) to determine the values of (a) where (f(a)) is strictly increasing:

The term (5a^4) is always non-negative for (a geq 0).

The term (-3a^2) is non-positive for (a geq 0).

Since (5a^4 - 3a^2) is strictly positive for all (a > 0), it follows that (f(a)) is strictly increasing for (a > 0).

Thus, there is exactly one root (a_0) in the interval ([1, 2]).

Step 3: Determining the Range of (a^6)

Since there is one unique positive root (a_0) in the interval ([1, 2]), we can find the corresponding range for (a^6):

For (a 1):

a^6 1^6 1.

For (a 2):

a^6 2^6 64.

Since (a) is strictly increasing and lies in the interval ([1, 2]), (a^6) will take values in the interval ([1, 64]).

Conclusion

Thus, the range of (a^6) is:

boxed{1, 64}.

Alternate Solution

Consider the equation (a^5 - a^3a 2). Multiplying both sides by (a^2), we get:

a^2(a^5 - a^3a) 2a^2.

This can be rewritten as:

a^7 - a^5a^3a^5 - a^3a 2a^2.

Dividing both sides by (a) (assuming (a eq 0)), we get:

a^6 frac{2a^2}{a^2 - 1}.

Now, we need to determine the possible values of (a):

If (a > 2), then (2a) will be greater than (2^2 4), and the denominator (a^2 - 1) will be greater than 3. Hence, the right-hand side will be less than (1), which is a contradiction since (a^6 > 1).

If (a

For (1

When (a 1):

a^6 1^6 1.

When (a 2):

a^6 2^6 64.

Hence, for values near 1 and 2, (a^6) can take values in the range ([3, 4]).

Therefore, the range of (a^6) is:

boxed{3, 4}.