Extrema shift 3
f ( x ) = a e x − x + b has two distinct roots x 1 , x 2 ( x 1 = x 2 ) .
For all a ∈ [ 0 , + ∞ ) , What is always true for f ′ ( 2 x 1 + x 2 ) ?
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f(x)=a.exp(x)-x+b. f'(x)=a.exp(x)-1, f"(x) =a.exp(x)>0 for all x, since a>=0. Thus f(x) has a minimum at f'(x) =0. The zeros of f(x) are at x1 and x2. Therefore f'(x) is negative between these two values. Since (x1+x2)/2 lies within these values, f'((x1+x2)/2)<0