Calculus

Question

if g is a differentiable function such that g(x) < 0 for all real numbers x and if f '(x) = (x^2 -4)g(x), which of the following is true? A. f has a relative maximum at x = -2 and a relative min at x = 2 B. f has a relative min at x = -2 and a relative max at x = 2 C. f has relative minima at x = -2 and x = 2 D. f has relative maxima at x = -2 and x =2 E. it cannot be determined if f has any relative extrema

Answer

<P>Hi Louise,<BR><BR>f(x) may be factored into (x-2)(x+2)g(x)&nbsp;&nbsp;<BR><BR>For all x &lt; -2, the first factor is negative, the second factor is negative, and the third factor is negative.&nbsp; Multiply three negative factors and you get a negative.&nbsp; So f(x) is decreasing in this region.&nbsp;&nbsp;<BR><BR>For x between -2 and +2, the first factor is negative, the second factor is positive, and the third factor is negative.&nbsp; Multiply these factors together and you get a positive.&nbsp; f(x) is increasing.&nbsp;&nbsp;<BR><BR>For x &gt; 2, the first factor is positive, the second factor is positive, and the third factor is negative.&nbsp; Multiply these factors together and you get a negative.&nbsp; f(x) is decreasing.&nbsp;&nbsp;<BR><BR>So what happens when f(x) is decreasing less than -2 and increasing greater than -2?&nbsp; You get a relative minimum at x = -2.&nbsp; What happens when f(x) is increasing less than +2 and then decreasing when x is greater than +2?&nbsp; You get a relative maximum at +2.&nbsp; It looks to me that (b) is the correct answer.<BR><BR><BR>Joe M.&nbsp;<BR>Info Tech Spec.<BR>FIA<BR>Knoxville, TN&nbsp; </P> http://vrd.askvrd.org/services/answerschema.xml