if ( f(4)=9 ) and ( f^{prime}(x) geq 2 ) for ( 4 leq x leq 7 ), how small can ( f(7) ) possibly be? ( f(7)…

Explanation:

Step1: Apply the Mean Value Theorem

The Mean Value Theorem states that if (y = f(x)) is continuous on ([a,b]) and differentiable on ((a,b)), then (f(b)-f(a)=f^{\prime}(c)(b - a)) for some (c\in(a,b)). Here, (a = 4), (b = 7), so (f(7)-f(4)=f^{\prime}(c)(7 - 4)=3f^{\prime}(c)), where (4\lt c\lt7).

Step2: Use the given condition (f^{\prime}(x)\geq2)

Since (f^{\prime}(c)\geq2) (because (f^{\prime}(x)\geq2) for (4\leq x\leq7)), then (f(7)-f(4)\geq3\times2).

Step3: Substitute (f(4) = 9)

We know (f(4) = 9). So (f(7)-9\geq6). Add 9 to both sides of the inequality: (f(7)\geq9 + 6).

Answer:

(15)