Question

Let f : [1,3] $\to$ R be a continuous function that is differentiable in (1, 3) an $f^{\prime }(x)=|f(x)|2+4$ for all x $\in$ (1,3). Then,

• A. f(3) - f(1) = 5 is true
• B. f(3) - f(1) = 5 is false
• C. f(3) - f(1) = 7 is false
• D. f(3) - f(1) < 0 only at one point of (1,3)

Answer 1

Answer: (B), (C)

By applying LMVT, there exist at least one c$\in$(1,3) such that $\frac{f\left(3\right)-f\left(1\right)}{3-1}=f^{\prime }\left(c\right)$
$\Rightarrow f\left(3\right)-f\left(1\right)=2.{\left|f\left(c\right)\right|}^2+8\Rightarrow f\left(3\right)-f\left(1\right)\ge 8$