Determine whether Rolle's Theorem applies to the following func $$ g(x)=x^{3}+10 x^{2}+32 x+32 ;[-4,-2] $$

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To determine whether **Rolle's Theorem** applies to the function $$ g(x) = x^{3} + 10x^{2} + 32x + 32 $$ on the interval $$[-4, -2]$$, we need to verify the three essential conditions of the theorem:

1. **Continuity on $$[a, b]$$:**
   - **Polynomial functions** like $$ g(x) $$ are continuous everywhere on $$\mathbb{R}$$. 
   - Therefore, $$ g(x) $$ is continuous on $$[-4, -2]$$.

2. **Differentiability on $$(a, b)$$:**
   - Polynomial functions are also differentiable everywhere on $$\mathbb{R}$$.
   - Hence, $$ g(x) $$ is differentiable on $$(-4, -2)$$.

3. **Equal Function Values at Endpoints ($$ g(a) = g(b) $$):**
   - Calculate $$ g(-4) $$ and $$ g(-2) $$:
     $$
     g(-4) = (-4)^3 + 10(-4)^2 + 32(-4) + 32 = -64 + 160 - 128 + 32 = 0
     $$
     $$
     g(-2) = (-2)^3 + 10(-2)^2 + 32(-2) + 32 = -8 + 40 - 64 + 32 = 0
     $$
   - Since $$ g(-4) = g(-2) = 0 $$, the third condition is satisfied.

**Conclusion:**
All three conditions of Rolle's Theorem are met for the function $$ g(x) $$ on the interval $$[-4, -2]$$. Therefore, **Rolle's Theorem does apply** to this function on the given interval.

**Implication:**
There exists at least one number $$ c $$ in $$(-4, -2)$$ such that $$ g'(c) = 0 $$.

Answer: Yes. All conditions of Rolle’s Theorem are met for g  on [–4, –2], so the theorem applies.
Rolle's Theorem applies to the function $$ g(x) = x^{3} + 10x^{2} + 32x + 32 $$ on the interval $$[-4, -2]$$ because $$ g(x) $$ is continuous and differentiable there, and $$ g(-4) = g(-2) = 0 $$.

Determine whether Rolle's Theorem applies to the following func \( g(x)=x^{3}+10 x^{2}+32 x+32 ;[-4,-2] \)

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