The function is concave down for x in what interval?

Question

The function

f(x)=x*sqrt(x^2+8)+9

is concave down for x in the interval_______ 

Answer 1

f(x) = x(x² + 8)^1/2 + 9

 

 

To the find the intervals of concavity, we set the second derivative equal to zero.  To find the second derivative, we derive f(x), then find the derivative of that derivative.

 

 

Be mindful of your algebra as you derive.

 

 

f'(x) = (x² + 8)^1/2 + x(1/2)(x² + 8)^-1/2(2x)

 

       = (x² + 8)^1/2 + x² (x² + 8)^-1/2

 

       = (x² + 8)^-1/2 [(x² + 8) + x²]

     

       = (x² + 8)^-1/2 (2x² + 8)

 

 

f''(x) = 0

 

(-1/2)(x² + 8)^-3/2 (2x)(2x² + 8) + (x² + 8)^-1/2 (4x) = 0

 

-x(x² + 8)^-3/2(2x² + 8) + (x² + 8)^-1/2(4x) = 0

 

x(x² + 8)^-3/2 [-1(2x² + 8) + 4(x² + 8)] = 0

 

x(x² + 8)^-3/2 (-2x² - 8 + 4x² + 32) = 0

 

x(x² + 8)^-3/2 (2x² + 24) = 0

 

 

Set the factors equal to zero.

 

x = 0        ,       1 / √[(x² + 8)³] = 0        ,        2(x² + 12) = 0

 

 

We discard the second and third factor, being that the second factor has a constant numerator of 1 and the third factor will give us a complex number.

 

 

x = 0  is our inflection point.  This is the location where f(x) changes concavity.

 

 

Evaluate f''(x) when x = -1 and x = 1 to determine where f(x) is concave down.

 

If f''(-1) is negative, then f(x) is concave down in the interval (-∞, 0].

If f''(1) is negative, then f(x) is convcave down in the interval [0, ∞).

 

 

 

Answer 2

To answer this question use a graphing calculator to graph the function.  when the function is curving downward it is concave down.  Therefore it is concaving down for x from( -∞, 0).