First slide
Theory of expressions

Let aR and f:RR be given by f(x)=x55x+a then,


Let y=x55x 

 (i) As x,y and x,y

(ii) Also, at x = 0, y = 0, thus the curve passes through the origin.

 (iii) dydx=5x45=5x41=5x21x2+1


Now, dydx>0 in (,1)(1,) thus f(x) is increasing in these interval

Also, dydx<0(1,1) thus decreasing in (-1, 1).

(iv) Also , at x = -1,  dydx changes its sign from + ve to ve

x= - 1 is point of local maxima.
Similarly, x = 1 is point of local minima

Local maximum value,y=(1)55(1)=4

Local minimum value,y=(1)55(1)=4

Now, let y = -a
As, evident from the graph, if a(4,4) i.e., a(4,4)

Then, f(x) has three real roots and

 if a>4 or a>4 then f(x) has one real root.

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