What are the critical values, if any, of # f(x)= (x-2)/(x^2-4)+2x#?

Question

Emily Ammerman · Accepted Answer

The critical points are:

#x= -2+- 1/sqrt(2)# We can simplify the function and remove one discontinuity noting that: #(x^2-4) = (x-2)(x+2)# hence: #f(x) = (x-2)/(x^2-4)+2x = (x-2)/((x-2)(x+2))+2x =1/(x+2) +2x# #f'(x) = -1/((x+2)^2)+2# So we can find the critical points as roots of the equation: #-1/((x+2)^2)+2 = 0# #1/((x+2)^2)= 2# #(x+2)^2 = 1/2# #x= -2+- 1/sqrt(2)# As: #f''(x) = 2/((x+2)^3)# we have: #f''(-2- 1/sqrt(2)) = 2/((-2- 1/sqrt(2)+2)^3)=-4sqrt(2) <0# #f''(-2+ 1/sqrt(2)) = 2/((-2+ 1/sqrt(2)+2)^3)=4sqrt(2) >0# so #x=-2- 1/sqrt(2)# is a local maximum and #x=-2+ 1/sqrt(2)# is a local minimum. graph{(x-2)/(x^2-4)+2x [-23.13, 16.87, -13.6, 6.4]}

Charles Arocha · Answer

undefined To find the critical values of the function $ f(x) = \frac{x - 2}{x^2 - 4} + 2x $, we need to first find the derivative of the function, set it equal to zero, and solve for $ x $. Then we check for any critical points that may arise.

The derivative of the function $ f(x) $ with respect to $ x $ is:

$$ f'(x) = \frac{d}{dx} \left( \frac{x - 2}{x^2 - 4} \right) + \frac{d}{dx}(2x) $$

$$ = \frac{(x^2 - 4) \cdot \frac{d}{dx}(x - 2) - (x - 2) \cdot \frac{d}{dx}(x^2 - 4)}{(x^2 - 4)^2} + 2 $$

$$ = \frac{(x^2 - 4) \cdot 1 - (x - 2) \cdot (2x)}{(x^2 - 4)^2} + 2 $$

$$ = \frac{x^2 - 4 - 2x(x - 2)}{(x^2 - 4)^2} + 2 $$

$$ = \frac{x^2 - 4 - 2x^2 + 4x}{(x^2 - 4)^2} + 2 $$

$$ = \frac{-x^2 + 4x - 4}{(x^2 - 4)^2} + 2 $$

Setting $ f'(x) $ equal to zero:

$$ \frac{-x^2 + 4x - 4}{(x^2 - 4)^2} + 2 = 0 $$

$$ \frac{-x^2 + 4x - 4 + 2(x^2 - 4)^2}{(x^2 - 4)^2} = 0 $$

$$ -x^2 + 4x - 4 + 2(x^2 - 4)^2 = 0 $$

$$ 2(x^2 - 4)^2 - x^2 + 4x - 4 = 0 $$

$$ 2(x^4 - 8x^2 + 16) - x^2 + 4x - 4 = 0 $$

$$ 2x^4 - 16x^2 + 32 - x^2 + 4x - 4 = 0 $$

$$ 2x^4 - 17x^2 + 4x + 28 = 0 $$

This is a quartic equation which may not have easily expressible solutions. You would typically use numerical methods to approximate the roots of this equation, which would then represent the critical values of the function.