Find the absolute maximum of `x^(40) -x^(20)` on the interval [0,1].

Answer

To find the absolute maximum of the function \( f(x) = x^{40} - x^{20} \) on the interval \([0, 1]\), we will follow these steps:

### Step 1: Evaluate the function at the endpoints of the interval
We need to check the values of the function at the endpoints \( x = 0 \) and \( x = 1 \).

\[ f(0) = 0^{40} - 0^{20} = 0 - 0 = 0 \]

\[ f(1) = 1^{40} - 1^{20} = 1 - 1 = 0 \]

### Step 2: Find the derivative of the function
Next, we find the derivative of the function \( f(x) \):

\[ f'(x) = \frac{d}{dx}(x^{40} - x^{20}) = 40x^{39} - 20x^{19} \]

### Step 3: Set the derivative equal to zero to find critical points
We set the derivative equal to zero to find critical points:

\[ 40x^{39} - 20x^{19} = 0 \]

Factoring out the common terms:

\[ 20x^{19}(2x^{20} - 1) = 0 \]

This gives us two factors:

1. \( 20x^{19} = 0 \) which gives \( x = 0 \)
2. \( 2x^{20} - 1 = 0 \) which gives \( x^{20} = \frac{1}{2} \)

From \( x^{20} = \frac{1}{2} \), we find:

\[ x = \left(\frac{1}{2}\right)^{\frac{1}{20}} \]

### Step 4: Evaluate the function at the critical point
Now we evaluate the function at the critical point \( x = \left(\frac{1}{2}\right)^{\frac{1}{20}} \):

\[ f\left(\left(\frac{1}{2}\right)^{\frac{1}{20}}\right) = \left(\left(\frac{1}{2}\right)^{\frac{1}{20}}\right)^{40} - \left(\left(\frac{1}{2}\right)^{\frac{1}{20}}\right)^{20} \]

Calculating each term:

\[ = \left(\frac{1}{2}\right)^{\frac{40}{20}} - \left(\frac{1}{2}\right)^{\frac{20}{20}} = \left(\frac{1}{2}\right)^{2} - \left(\frac{1}{2}\right)^{1} = \frac{1}{4} - \frac{1}{2} \]

\[ = \frac{1}{4} - \frac{2}{4} = -\frac{1}{4} \]

### Step 5: Compare the values to find the absolute maximum
Now we compare the values of \( f(0) \), \( f(1) \), and \( f\left(\left(\frac{1}{2}\right)^{\frac{1}{20}}\right) \):

- \( f(0) = 0 \)
- \( f(1) = 0 \)
- \( f\left(\left(\frac{1}{2}\right)^{\frac{1}{20}}\right) = -\frac{1}{4} \)

The maximum value among these is \( 0 \).

### Conclusion
Thus, the absolute maximum of \( f(x) \) on the interval \([0, 1]\) is:

\[ \boxed{0} \]