Question

Find the maximum and minimum values of the function below on the horizontal span from 1 to 5. Be sure to include endpoint maxima or minima. (Round your answers to two decimal places.) x^2 + 85/x

Answer 1

Answer:

Max = 86; min = 36.54

Step-by-step explanation:

$f(x) = x^{2} + \dfrac{85}{x}$

Step 1. Find the critical points.

(a) Take the derivative of the function.

$f'(x) = 2x - \dfrac{85}{x^{2}}$

Set it to zero and solve.

$\begin{array}{rcl}2x - \dfrac{85}{x^{2}} & = & 0\\\\2x^{3} - 85 & = & 0\\2x^{3} & = & 85\\\\x^{3} & = &\dfrac{85}{2}\\\\x & = & \sqrt [3]{\dfrac{85}{2}}\\\\& \approx & 3.490\\\end{array}$

(b) Calculate ƒ(x) at the critical point.  

$f(3.490) = 3.490^{2} + \dfrac{85}{3.490} = 12.18 + 24.36 = 36.54$

Step 2. Calculate ƒ(x) at the endpoints of the interval

$f(1) = 1^{2} + \dfrac{85}{1} = 1 + 85 = 86\\\\f(5) = 5^{2} + \dfrac{85}{5} = 25 + 17 = 42$

Step 3.Identify the maxima and minima.

ƒ(x) achieves its absolute maximum of 86 at x = 1 and its absolute minimum of 36.54 at x = 3.490

The figure below shows the graph of ƒ(x) from x = 1 to x = 5.