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Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Check Rolle's theorem and find its interior point.
- Justify the conclusion "f′(x)=2x−4=0 at c=2" using the stated assumptions.
Before you start
Continuity and differentiation.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Apply Rolle's theorem to f(x)=x²−4x+3 on [1,3].
Why this math matters
Check Rolle's theorem and find its interior point. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The interval has distinct endpoints.
- All theorem hypotheses are checked rather than inferred from a sketch.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Guarantee a horizontal tangent between equal endpoint heights
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Apply Rolle's theorem to f(x)=x²−4x+3 on [1,3].
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
f(1)=f(3)=0
The endpoints share a height.
Work through the mathematics
The polynomial is continuous on [1,3] and differentiable on (1,3)
All hypotheses are satisfied.
Check the conclusion
f′(x)=2x−4=0 at c=2
The theorem guarantees a horizontal tangent somewhere inside; solving finds it here.
The result
f′(x)=2x−4=0 at c=2
The theorem guarantees a horizontal tangent somewhere inside; solving finds it here.
Common mistakes to catch
- Rolle's theorem concerns an interior point.
- Dropping differentiability can invalidate the conclusion.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Would equal endpoint values alone be sufficient?
Show a hint
A corner can defeat differentiability.
Reveal answer and explanation
No
For example, an absolute-value graph can meet equal heights while lacking an interior zero derivative.
Practice 2
Does the theorem require the endpoints to have height zero?
Show a hint
It requires equality, not a particular value.
Reveal answer and explanation
No
Any common endpoint value is allowed.
Take the idea with you
Use endpoint agreement as a reason to search for an intermediate stationary state.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Recognize a horizontal tangent without a maximum or minimum
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