Current location - Training Enrollment Network - Mathematics courses - mathematical problem
mathematical problem
According to the increase or decrease.

f(x)=xe^ax- 1/x,x≠0

f(x)=x(e^ax- 1/x? )x-& gt; 0+,f(x)-& gt; -∞; x-& gt; 0-,f(x)-& gt; +∞, x=0 is a breakpoint, but not a zero point.

F(x) has two zeros, which is equivalent to g (x) = e ax- 1/x? There are two zeros.

For example, y = (e a) x and y= 1/x? Intersection point, as shown in the figure below:

A = 0,e 0 = 1,y = (e A) x = 1 x = 1,y= 1/x? There is an intersection on each side;

e^a>; 1=e^0,a>; 0 and one

As shown above, there must be an intersection point on the right side of the Y axis;

On the left side of the y axis, x

When tangent, the slopes of two curves are equal at the tangent point. Tangent points are also intersections.

Let the intersection (x0, y0), x0.

y 1=e^ax,y 1'=ae^x,>; 0y0=e^ax0

y2= 1/x? ,y2'=-2/x? & gt0,

ae^x0=-2/x0? ;

y0=e^ax0

y0= 1/x0?

ay0=-2/x0? ;

a=(-2/x0? )/( 1/x0? )=-2x0,x0=-a/2

y0= 1/(a? /4)=4/a?

ay0=-2/(-a? /8)= 16/a? ,y0= 16/a^4

4/a? = 16/a^4

1=4/a?

Answer? =4,a=2

∴-2≤a≤2