(Solved): I need help with the last one c= everything else is right The goal of this problem is to show that t ...

The goal of this problem is to show that the function $f(x)=x+\frac{2}{x}$ satisfies both of the conditions (the hypotheses) and the conclusion of the Mean Value Theorem for $x$ in the interval $\left[\frac{1}{4},4\right]$. Verification of Hypotheses: Fill in the blanks to show that the hypotheses of the Mean Value Theorem are satisfied: $f(x)$ is $E$ on $\left[\frac{1}{4},4\right]$ and is on $\left(\frac{1}{4},4\right)$. Note: The answer in each box should be one word. Verification of the Conclusion: If the hypotheses of the Mean Value Theorem are satisfied, then there is at least one $c$ in the interval $\left(\frac{1}{4},4\right)$ $f^{\prime }(c)=\frac{f(4)-f\left(\frac{1}{4}\right)}{4-\frac{1}{4}}.$ Verify that the conclusion of the Mean Value Theorem holds by computing $\frac{f(4)-f\left(\frac{1}{4}\right)}{4-\frac{1}{4}}=$ Now find $c$ in $\left(\frac{1}{4},4\right)$ so that $f^{\prime }(c)$ equals the answer you just found. (For this problem there is only one correct value of $c$.) $c=$