Q.11 - Calculus

Question

We could use part (c) from Theorem 3.6 to add a third part to Theorem 3.10 that would tell us what it means when f'' is zero in the interior of an interval I. Fill in the blank accordingly: If f'' is zero on I, then f is             on I. 

Step-by-Step Solution

Answer

If f'' is zero on I, then f is constant on I. 

Step 1. Given information.

The given incomplete statement is the following.

If f'' is zero on I, then f is             on I. 

Step 2. Complete statement.

Theorem $3.6$The Derivative Measures Where a Function is Increasing or Decreasing states that the f be a function that is differentiable on an interval I than

(a) If f' is positive in the interior of I, then f is increasing on I.

(b) If f' is negative in the interior of I, then f is decreasing on I.

(c) If f' is zero in the interior of I, then f is constant on I.

Using part (c), the theorem $3.10$would be following.

If f'' is zero on I, then f is constant on I.