Using graphs to understand a question.
The curve has a local minimum at . How many real solutions are there to the equation ?
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Factorising f(x) gives f ( x ) = x ( x 4 − 2 x 2 + 1 ) = x ( x 2 − 1 ) 2 = x ( x − 1 ) 2 ( x + 1 ) 2
By considering the curve f(x-1), the equation becomes f ( x − 1 ) = ( x − 1 ) 5 − 2 ( x − 1 ) 3 + x − 1 = ( x − 1 ) ( x − 2 ) 2 ( x ) 2 = x 5 − 5 x 4 + 8 x 3 − 4 x 2 . This is a translation of the above curve right one unit. By adding one (Translating up one unit), we get the equation in the question. The new position of the turning point in the question is now (0.553, 0.714) which is above the x axis. Using the graph shows that there is only one solution when f(x-1) + 1 = 0.