1. The problem asks for the second derivative of the function $$f(x) = x^7 + x^3 - 21$$ evaluated at $$x=1$$. 2. Recall the rules for derivatives: - The derivative of $$x^n$$ is $$nx^{n-1}$$. - The second derivative is the derivative of the first derivative. 3. First, find the first derivative: $$f'(x) = \frac{d}{dx}(x^7) + \frac{d}{dx}(x^3) - \frac{d}{dx}(21) = 7x^6 + 3x^2 - 0 = 7x^6 + 3x^2$$ 4. Next, find the second derivative: $$f''(x) = \frac{d}{dx}(7x^6) + \frac{d}{dx}(3x^2) = 7 \cdot 6x^{5} + 3 \cdot 2x^{1} = 42x^{5} + 6x$$ 5. Evaluate the second derivative at $$x=1$$: $$f''(1) = 42(1)^5 + 6(1) = 42 + 6 = 48$$ 6. Therefore, the second derivative of the function at $$x=1$$ is **48**.