### Video Transcript

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

To use the mean value theorem, two
things must be true about our function π of π₯. It must be continuous over the
closed interval π to π. And it must be differentiable over
the open interval π to π. Well, the function two π₯ cubed
minus four π₯ plus seven is indeed continuous over the closed interval zero to
five. Itβs a simple cubic graph that
looks a little like this over our closed interval. And to check for the second
condition, weβll see what happens when we do differentiate with respect to π₯. The derivative of two π₯ cubed is
three times two π₯ squared. Thatβs six π₯ squared. And the derivative of negative four
π₯ is negative four. So we obtain dπ¦ by dπ₯ to be equal
to six π₯ 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.