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.