Q. 6 - Calculus

Question

State the contrapositive of Theorem 3.10(a). Is the contrapositive true? If so, explain why; if not, provide a counterexample. 

Step-by-Step Solution

Answer

The contrapositive of Theorem state that if f and f' are both differentiable on an interval I and f is not concave up on I  then f''  will not be positive on I .

The contrapositive theorem of $3.10\left(a\right)$is true because the second derivative might be zero at some point.

Step 1. Given information.

The given theorem is as follows.

Suppose f and f' are both differentiable on an interval I.

If f'' is positive on I, then f is concave up on I.

Step 2. Statement of the contrapositive of Theorem.

The contrapositive of Theorem state that if f and f' are both differentiable on an interval I and f is not concave up on I  then f''  will not be positive on I .

Step 3. An explanation for contrapositive of Theorem.

If the second derivative f'' of a function f is Positive then f'  is increasing on the interval, and if f' is increasing then f will concave up on interval.

Therefore f is concave up or down can be checked by the sign of its second derivative.

So the contrapositive theorem of $3.10\left(a\right)$is true because the second derivative might be zero at some point.