Question Video: The Mean Value Theorem Mathematics • Higher Education

Does the mean value theorem apply for the function 𝑦 = 2𝑥³ − 4𝑥 + 7 over the closed interval [0, 5]?

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Video Transcript

Does the mean value theorem apply for the function 𝑦 equals two 𝑥 cubed minus four 𝑥 plus seven over the closed interval zero to five?

To use the mean value theorem, two things must be true about our function 𝑓 of 𝑥. It must be continuous over the closed interval 𝑎 to 𝑏. And it must be differentiable over the open interval 𝑎 to 𝑏. Well, the function two 𝑥 cubed minus four 𝑥 plus seven is indeed continuous over the closed interval zero to five. It’s a simple cubic graph that looks a little like this over our closed interval. And to check for the second condition, we’ll see what happens when we do differentiate with respect to 𝑥. The derivative of two 𝑥 cubed is three times two 𝑥 squared. That’s six 𝑥 squared. And the derivative of negative four 𝑥 is negative four. So we obtain d𝑦 by d𝑥 to be equal to six 𝑥 squared minus four. This is indeed defined over the open interval zero to five. And we can say yes, the mean value theorem does indeed apply.

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