Calculations:

• First derivative: $f'(x) = 3x^2 + 2ax + b$.
• Since relative extrema occur at critical points $x = -3$ and $x = 1$, these are the roots of $f'(x) = 0$:
  • Sum of roots: $(-3) + 1 = -\frac{2a}{3} \implies -2 = -\frac{2a}{3} \implies a = 3$

• Product of roots: $(-3)(1) = \frac{b}{3} \implies -3 = \frac{b}{3} \implies b = -9$

• To find $c$, use the minimum point $(1, -14)$:
  $$f(1) = 1^3 + 3(1)^2 - 9(1) + c = -14$$
  $$1 + 3 - 9 + c = -14 \implies -5 + c = -14 \implies c = -9$$
• Verification at $(-3, 18)$:
  $$f(-3) = (-27) + 3(9) - 9(-3) - 9 = -27 + 27 + 27 - 9 = 18$$

A function is concave up on intervals where its second derivative is strictly positive ($f''(x) \gt 0$):

• First derivative:
  $$f'(x) = 4x^3 - 12x^2$$
• Second derivative:
  $$f''(x) = 12x^2 - 24x = 12x(x - 2)$$
• Set $f''(x) \gt 0$:
  $$12x(x - 2) \gt 0$$
  The roots are $x = 0$ and $x = 2$. The product is positive when:
  $x \lt 0$ or $x \gt 2$
  In interval notation, the graph is concave up on $(-\infty, 0) \cup (2, \infty)$.

This is the exact, classical statement of Rolle's Theorem:
If a real-valued function $f$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c \in (a, b)$ such that $f'(c) = 0$.

Detailed analysis of options:

• Lagrange's Mean Value Theorem states that $f'(c) = \frac{f(b) - f(a)}{b - a}$ (without requiring $f(a) = f(b)$).
• Rolle's Theorem is the specific case where $f(a) = f(b)$, implying $f'(c) = 0$.
• Fundamental Theorem of Calculus relates differentiation and integration.
• Fundamental Theorem of Algebra states that every non-zero single-variable polynomial with complex coefficients has at least one complex root.

A point $(x_0, f(x_0))$ on a curve where the graph changes from concave upward to concave downward, or from concave downward to concave upward, is called a point of inflection (inflection point).

Detailed analysis of options:

• relative maximum occurs where $f'$ changes from positive to negative, not merely where concavity changes.
• relative minimum occurs where $f'$ changes from negative to positive.
• An inflection point at $x_0$ is the precise mathematical term for a change in concavity.
• saddle point applies to multivariable surfaces, not single-variable curves.