Question
What is the concavity of the quadratic function f(x) = x^2 + 5x?
Answers
GPT-4o mini
To determine the concavity of the quadratic function \( f(x) = x^2 + 5x \), we can find the second derivative of the function.
1. **First derivative**: We start by calculating the first derivative \( f'(x) \):
\[
f'(x) = \frac{d}{dx}(x^2 + 5x) = 2x + 5.
\]
2. **Second derivative**: Now, we find the second derivative \( f''(x) \):
\[
f''(x) = \frac{d}{dx}(2x + 5) = 2.
\]
The second derivative \( f''(x) = 2 \) is a constant and is positive.
Since the second derivative is positive, this means that the function \( f(x) = x^2 + 5x \) is concave up for all values of \( x \).
Thus, the answer is that the function is concave up everywhere.
1. **First derivative**: We start by calculating the first derivative \( f'(x) \):
\[
f'(x) = \frac{d}{dx}(x^2 + 5x) = 2x + 5.
\]
2. **Second derivative**: Now, we find the second derivative \( f''(x) \):
\[
f''(x) = \frac{d}{dx}(2x + 5) = 2.
\]
The second derivative \( f''(x) = 2 \) is a constant and is positive.
Since the second derivative is positive, this means that the function \( f(x) = x^2 + 5x \) is concave up for all values of \( x \).
Thus, the answer is that the function is concave up everywhere.