Find the critical (stationary ) points of the function f(X)=`(x^(5))/(20)-(x^(4))/(12)`+5Name these points .Also find the point of inflection

Explanation

To solve the problem of finding the critical points and points of inflection for the function \( f(x) = \frac{x^5}{20} - \frac{x^4}{12} + 5 \), we will follow these steps:

### Step 1: Find the first derivative \( f'(x) \)

To find the critical points, we first need to compute the first derivative of the function:

\[ f'(x) = \frac{d}{dx} \left( \frac{x^5}{20} - \frac{x^4}{12} + 5 \right) \]

Using the power rule of differentiation, we get:

\[ f'(x) = \frac{5x^4}{20} - \frac{4x^3}{12} \]

Simplifying this, we have:

\[ f'(x) = \frac{x^4}{4} - \frac{x^3}{3} \]

### Step 2: Set the first derivative to zero

To find the critical points, we set \( f'(x) = 0 \):

\[ \frac{x^4}{4} - \frac{x^3}{3} = 0 \]

To solve this equation, we can factor out \( x^3 \):

\[ x^3 \left( \frac{x}{4} - \frac{1}{3} \right) = 0 \]

This gives us two factors to consider:

1. \( x^3 = 0 \) which leads to \( x = 0 \)
2. \( \frac{x}{4} - \frac{1}{3} = 0 \)

Solving the second equation:

\[ \frac{x}{4} = \frac{1}{3} \implies x = \frac{4}{3} \]

### Step 3: Identify the critical points

From the calculations, we find the critical points to be:

- \( x = 0 \)
- \( x = \frac{4}{3} \)

### Step 4: Determine the nature of the critical points

To determine whether these points are maxima or minima, we can use the first derivative test or find the second derivative.

### Step 5: Find the second derivative \( f''(x) \)

We differentiate \( f'(x) \) to find \( f''(x) \):

\[ f''(x) = \frac{d}{dx} \left( \frac{x^4}{4} - \frac{x^3}{3} \right) \]

Calculating this gives:

\[ f''(x) = \frac{4x^3}{4} - \frac{3x^2}{3} = x^3 - x^2 \]

### Step 6: Set the second derivative to zero for points of inflection

To find the points of inflection, we set \( f''(x) = 0 \):

\[ x^3 - x^2 = 0 \]

Factoring out \( x^2 \):

\[ x^2 (x - 1) = 0 \]

This gives us:

1. \( x^2 = 0 \) which leads to \( x = 0 \)
2. \( x - 1 = 0 \) which leads to \( x = 1 \)

### Summary of Results

- **Critical Points**: \( x = 0 \) (local maximum) and \( x = \frac{4}{3} \) (local minimum).
- **Points of Inflection**: \( x = 0 \) and \( x = 1 \).