At the point with x-coordinate $-2$, is the graph of $g\left(x\right)=4x^3+9x^2+7x+3$ concave up or concave down?

• A. Concave up
• B. Concave down
• C. Neither (the concavity is undefined at this point)
• D.

Answer: (B)

Verified step by step guidance

1. Step 1: Recall that the concavity of a function is determined by the sign of its second derivative. If the second derivative is positive at a given point, the graph is concave up. If the second derivative is negative, the graph is concave down.
2. Step 2: Compute the first derivative of g(x) = 4x^3 + 9x^2 + 7x + 3. Use the power rule: \( g'(x) = \frac{d}{dx}(4x^3) + \frac{d}{dx}(9x^2) + \frac{d}{dx}(7x) + \frac{d}{dx}(3) \).
3. Step 3: Simplify the first derivative: \( g'(x) = 12x^2 + 18x + 7 \).
4. Step 4: Compute the second derivative of g(x) by differentiating \( g'(x) \): \( g''(x) = \frac{d}{dx}(12x^2) + \frac{d}{dx}(18x) + \frac{d}{dx}(7) \).
5. Step 5: Evaluate \( g''(x) \) at \( x = -2 \). Substitute \( x = -2 \) into the second derivative and determine its sign. If \( g''(-2) > 0 \), the graph is concave up. If \( g''(-2) < 0 \), the graph is concave down.