Question

if we apply rolles theorem to the function (f(x) = 2x^2 - 16x - 1) on the interval (2, 6), how many values of (c) exist such that (f(c) = 0)?

what is the value of (c)?

if we try to apply rolles thorem to the function (f(x) = 2x^2 - 16x - 1) on the interval (-1, 13), which of the following conditions is not met?

(f(a) = f(b))
continuty on (-1, 13)
differentiability on (-1, 13)

Explanation:

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"current_concepts": [
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"Critical Points"
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</pre_analysis>

<reasoning>

Verify Rolle's Theorem conditions on [2, 6]

\[

$$\begin{aligned} &f(x) = 2x^2 - 16x - 1\\ &f(2) = 2(2)^2 - 16(2) - 1 = 8 - 32 - 1 = -25\\ &f(6) = 2(6)^2 - 16(6) - 1 = 72 - 96 - 1 = -25\\ &f(2) = f(6) = -25 \end{aligned}$$

\]

Solve for c in (2, 6) where f'(c) = 0

\[

$$\begin{aligned} &f'(x) = 4x - 16\\ &4c - 16 = 0 \implies c = 4\\ &c = 4 \in (2, 6) \end{aligned}$$

\]

Evaluate Rolle's Theorem conditions on [-1, 13]

\[

$$\begin{aligned} &f(-1) = 2(-1)^2 - 16(-1) - 1 = 2 + 16 - 1 = 17\\ &f(13) = 2(13)^2 - 16(13) - 1 = 338 - 208 - 1 = 129\\ &f(-1) eq f(13) \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

1

Question 2

4

Question 3

<mcq-correct>\(f(a) = f(b)\)</mcq-correct>
<mcq-option>continuity on \([-1, 13]\)</mcq-option>
<mcq-option>differentiability on \([-1, 13]\)</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Calculus",
"Rolle's Theorem"
]
}
</post_analysis>

Answer:

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"question_count": 3,
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"step_cot",
"explain_then_answer",
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],
"needs_drawing": false,
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"new_concepts": [
"Rolle's Theorem"
],
"current_concepts": [
"Rolle's Theorem",
"Derivative Definition",
"Critical Points"
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}
</pre_analysis>

<reasoning>

Verify Rolle's Theorem conditions on [2, 6]

\[

$$\begin{aligned} &f(x) = 2x^2 - 16x - 1\\ &f(2) = 2(2)^2 - 16(2) - 1 = 8 - 32 - 1 = -25\\ &f(6) = 2(6)^2 - 16(6) - 1 = 72 - 96 - 1 = -25\\ &f(2) = f(6) = -25 \end{aligned}$$

\]

Solve for c in (2, 6) where f'(c) = 0

\[

$$\begin{aligned} &f'(x) = 4x - 16\\ &4c - 16 = 0 \implies c = 4\\ &c = 4 \in (2, 6) \end{aligned}$$

\]

Evaluate Rolle's Theorem conditions on [-1, 13]

\[

$$\begin{aligned} &f(-1) = 2(-1)^2 - 16(-1) - 1 = 2 + 16 - 1 = 17\\ &f(13) = 2(13)^2 - 16(13) - 1 = 338 - 208 - 1 = 129\\ &f(-1) eq f(13) \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

1

Question 2

4

Question 3

<mcq-correct>\(f(a) = f(b)\)</mcq-correct>
<mcq-option>continuity on \([-1, 13]\)</mcq-option>
<mcq-option>differentiability on \([-1, 13]\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Calculus",
"Rolle's Theorem"
]
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