The quadratic approximation of f(x) = x3 – 3×2 – 5 a the point x = 0 is A. 3×2 – 6x – 5 B. -3×2 – 5 C. -3×2 + 6x – 5 D. 3×2 – 5

The quadratic approximation of $f(x) = x^3 – 3x^2 – 5$ at the point $x = 0$ is $3x^2 – 5$. This is because the quadratic polynomial $3x^2 – 5$ has the same first two derivatives as $f(x)$ at $x = 0$.

The first derivative of $f(x)$ is $3x^2 – 6x$. The second derivative of $f(x)$ is $6x$. The first derivative of $3x^2 – 5$ is $6x$. The second derivative of $3x^2 – 5$ is $0$.

Therefore, the quadratic approximation of $f(x)$ at the point $x = 0$ is $3x^2 – 5$.

Here is a more detailed explanation of each option:

• Option A: $3x^2 – 6x – 5$. This is the quadratic polynomial that has the same first two derivatives as $f(x)$ at $x = 0$. However, it does not pass through the point $(0, -5)$.
• Option B: $-3x^2 – 5$. This is the quadratic polynomial that passes through the point $(0, -5)$. However, it does not have the same first two derivatives as $f(x)$ at $x = 0$.
• Option C: $-3x^2 + 6x – 5$. This is the quadratic polynomial that has the same first derivative as $f(x)$ at $x = 0$. However, it does not have the same second derivative as $f(x)$ at $x = 0$.
• Option D: $3x^2 – 5$. This is the quadratic polynomial that has the same first two derivatives as $f(x)$ at $x = 0$ and passes through the point $(0, -5)$. Therefore, it is the quadratic approximation of $f(x)$ at the point $x = 0$.