The point of inflection of `y= x^(3) -5x^(2) + 3x-5` is

Options

A. `(1)/(2)`
B. `(3)/(4)`
C. `(7)/(4)`
D. `(5)/(3)`

Answer

To find the point of inflection of the function \( y = x^3 - 5x^2 + 3x - 5 \), we need to follow these steps:

### Step 1: Find the first derivative \( y' \)
The first derivative of \( y \) is calculated using the power rule of differentiation.

\[ y' = \frac{d}{dx}(x^3) - \frac{d}{dx}(5x^2) + \frac{d}{dx}(3x) - \frac{d}{dx}(5) \]

Calculating each term:

- The derivative of \( x^3 \) is \( 3x^2 \).
- The derivative of \( 5x^2 \) is \( 10x \).
- The derivative of \( 3x \) is \( 3 \).
- The derivative of a constant \( -5 \) is \( 0 \).

Putting it all together:

\[ y' = 3x^2 - 10x + 3 \]

### Step 2: Find the second derivative \( y'' \)
Now, we differentiate \( y' \) to find the second derivative.

\[ y'' = \frac{d}{dx}(3x^2) - \frac{d}{dx}(10x) + \frac{d}{dx}(3) \]

Calculating each term:

- The derivative of \( 3x^2 \) is \( 6x \).
- The derivative of \( 10x \) is \( 10 \).
- The derivative of a constant \( 3 \) is \( 0 \).

Putting it all together:

\[ y'' = 6x - 10 \]

### Step 3: Set the second derivative equal to zero
To find the points of inflection, we set the second derivative equal to zero:

\[ 6x - 10 = 0 \]

### Step 4: Solve for \( x \)
Now, we solve for \( x \):

\[ 6x = 10 \\ x = \frac{10}{6} = \frac{5}{3} \]

### Step 5: Find the corresponding \( y \) value
Now we substitute \( x = \frac{5}{3} \) back into the original function \( y \) to find the corresponding \( y \) value.

\[ y = \left(\frac{5}{3}\right)^3 - 5\left(\frac{5}{3}\right)^2 + 3\left(\frac{5}{3}\right) - 5 \]

Calculating each term:

- \( \left(\frac{5}{3}\right)^3 = \frac{125}{27} \)
- \( 5\left(\frac{5}{3}\right)^2 = 5 \cdot \frac{25}{9} = \frac{125}{9} \)
- \( 3\left(\frac{5}{3}\right) = 5 \)
- \( -5 = -5 \)

Now substituting these values into the equation:

\[ y = \frac{125}{27} - \frac{125}{9} + 5 - 5 \]

To combine these fractions, we need a common denominator, which is 27:

\[ y = \frac{125}{27} - \frac{375}{27} + \frac{135}{27} - \frac{135}{27} \\ y = \frac{125 - 375 + 135 - 135}{27} = \frac{-250}{27} \]

### Final Result
The point of inflection is:

\[ \left(\frac{5}{3}, -\frac{250}{27}\right) \]