`f(x)=x^(3)-3x+5,f(1.99)` is equal to

Options

A. 6.91
B. 9.19
C. 9.06
D. None of these

Explanation

To find the value of \( f(1.99) \) for the function \( f(x) = x^3 - 3x + 5 \), we can use the concept of derivatives and linear approximation.

### Step 1: Calculate \( f(2) \)

First, we compute \( f(2) \):
\[ f(2) = 2^3 - 3 \cdot 2 + 5 \]
Calculating this:
\[ f(2) = 8 - 6 + 5 = 7 \]

### Step 2: Find the derivative \( f'(x) \)

Next, we need to find the derivative \( f'(x) \):
\[ f'(x) = \frac{d}{dx}(x^3 - 3x + 5) = 3x^2 - 3 \]

### Step 3: Calculate \( f'(2) \)

Now, we evaluate the derivative at \( x = 2 \):
\[ f'(2) = 3 \cdot 2^2 - 3 = 3 \cdot 4 - 3 = 12 - 3 = 9 \]

### Step 4: Use linear approximation to find \( f(1.99) \)

Using the linear approximation formula:
\[ f(x + h) \approx f(x) + f'(x) \cdot h \]
Here, we set \( x = 2 \) and \( h = -0.01 \) (since \( 1.99 = 2 - 0.01 \)):
\[ f(1.99) \approx f(2) + f'(2) \cdot (-0.01) \]
Substituting the values we calculated:
\[ f(1.99) \approx 7 + 9 \cdot (-0.01) \]
\[ f(1.99) \approx 7 - 0.09 = 6.91 \]

### Final Answer
Thus, the approximate value of \( f(1.99) \) is:
\[ \boxed{6.91} \]