6. - / 2 points details my notes ask your teacher suppose that ( 3 leq f^{prime}(x) leq 5 ) for all values…

6. - / 2 points details my notes ask your teacher suppose that ( 3 leq f^{prime}(x) leq 5 ) for all values of ( x ). what are the minimum and maximum possible values of ( f(8)-f(3) )? ( leq f(8)-f(3) leq ) resources read it master it

6. - / 2 points details my notes ask your teacher suppose that ( 3 leq f^{prime}(x) leq 5 ) for all values of ( x ). what are the minimum and maximum possible values of ( f(8)-f(3) )? ( leq f(8)-f(3) leq ) resources read it master it

Answer

Explanation:

Step1: Apply the Mean Value Theorem

By the Mean Value Theorem, (f(8)-f(3)=f^{\prime}(c)(8 - 3)=5f^{\prime}(c)) for some (c\in(3,8)).

Step2: Use the given inequality for (f^{\prime}(x))

Since (3\leq f^{\prime}(x)\leq5), substitute (f^{\prime}(c)) into the inequality. Multiply each part of the inequality (3\leq f^{\prime}(c)\leq5) by (5). We get (3\times5\leq5f^{\prime}(c)\leq5\times5).

Answer:

(15\leq f(8)-f(3)\leq25)