Completing the Square?

Question

By completing the square on the denominator, find he maximum value of the function f(x)=6/x^2+6x+11. What value of x gives this maximum value.

Answer 1

The function is .

Apply first derivative with respect to x.

Apply second derivative with respect to x.

To find the critical numbers or does not exist.

Find the critical numbers .

Find the critical numbers, does not exist that means denominator not equals to zero..

Completing the square Method :

The denominator is (x² + 6x + 11)² = 0.

x² + 6x + 11 = 0

Separate variables and constants aside by subtracting 11 from each side.

x² + 6x = - 11

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

Here x coefficient = 6. So, (half the x coefficient)² = (6/2)²= 9.

Add 9 to each side.

x² + 6x + 9 = - 11 + 9

x² + 2(x)(3) + 3² = - 2

Apply Perfect Square Trinomial : u² + 2uv + v² = (u + v)².

(x + 3)² = (i√2)²                                    [ i² = - 1]

x + 3 = ± i√2

x = - 3 ± i√2.

The above imaginary numbers are not a critical numbers.

The critical number of the function is x = - 3.

If f " (c) > 0 (positive) ------> minimum point.

If f " (c) < 0 (negative) ------> maximum point.

To find the maximum point substitute in the original function.

The maximum point f(-3) = 3.