Find the absolute maximum and minimum values of each of the following in the given intervals :
`f(x)=x^(50)-x^(20),[0,1]`

Answer

To find the absolute maximum and minimum values of the function \( f(x) = x^{50} - x^{20} \) over the interval \([0, 1]\), we will follow these steps:

### Step 1: Find the derivative of the function
First, we need to differentiate the function \( f(x) \).

\[ f'(x) = \frac{d}{dx}(x^{50}) - \frac{d}{dx}(x^{20}) = 50x^{49} - 20x^{19} \]

### Step 2: Set the derivative to zero
Next, we set the derivative equal to zero to find the critical points.

\[ 50x^{49} - 20x^{19} = 0 \]

### Step 3: Factor the derivative
We can factor out the common terms from the equation:

\[ 10x^{19}(5x^{30} - 2) = 0 \]

### Step 4: Solve for critical points
Setting each factor to zero gives us:

1. \( 10x^{19} = 0 \) → \( x = 0 \)
2. \( 5x^{30} - 2 = 0 \) → \( 5x^{30} = 2 \) → \( x^{30} = \frac{2}{5} \) → \( x = \left(\frac{2}{5}\right)^{\frac{1}{30}} \)

### Step 5: Evaluate the function at critical points and endpoints
Now we evaluate \( f(x) \) at the critical points and the endpoints of the interval \([0, 1]\).

1. At \( x = 0 \):
\[ f(0) = 0^{50} - 0^{20} = 0 \]

2. At \( x = 1 \):
\[ f(1) = 1^{50} - 1^{20} = 1 - 1 = 0 \]

3. At \( x = \left(\frac{2}{5}\right)^{\frac{1}{30}} \):
\[ f\left(\left(\frac{2}{5}\right)^{\frac{1}{30}}\right) = \left(\frac{2}{5}\right)^{\frac{50}{30}} - \left(\frac{2}{5}\right)^{\frac{20}{30}} = \left(\frac{2}{5}\right)^{\frac{5}{3}} - \left(\frac{2}{5}\right)^{\frac{2}{3}} \]
Let \( y = \left(\frac{2}{5}\right)^{\frac{1}{3}} \), then:
\[ f\left(\left(\frac{2}{5}\right)^{\frac{1}{30}}\right) = y^5 - y^2 \]

### Step 6: Compare values
Now we need to compare the values at \( x = 0 \), \( x = 1 \), and \( x = \left(\frac{2}{5}\right)^{\frac{1}{30}} \).

1. \( f(0) = 0 \)
2. \( f(1) = 0 \)
3. \( f\left(\left(\frac{2}{5}\right)^{\frac{1}{30}}\right) = \left(\frac{2}{5}\right)^{\frac{5}{3}} - \left(\frac{2}{5}\right)^{\frac{2}{3}} \)

To determine whether \( f\left(\left(\frac{2}{5}\right)^{\frac{1}{30}}\right) \) is positive or negative, we can analyze the expression:
- Since \( \left(\frac{2}{5}\right)^{\frac{5}{3}} < \left(\frac{2}{5}\right)^{\frac{2}{3}} \) (as \( \frac{2}{5} < 1 \)), it follows that \( f\left(\left(\frac{2}{5}\right)^{\frac{1}{30}}\right) < 0 \).

### Conclusion
Thus, the absolute maximum value of \( f(x) \) on the interval \([0, 1]\) is \( 0 \) (occurring at both endpoints \( x = 0 \) and \( x = 1 \)), and the absolute minimum value is \( f\left(\left(\frac{2}{5}\right)^{\frac{1}{30}}\right) \), which is negative.

### Final Result
- Absolute Maximum: \( 0 \)
- Absolute Minimum: \( f\left(\left(\frac{2}{5}\right)^{\frac{1}{30}}\right) \)