= Solution
For a line $p(x)=ax+b$, the error $e^x-ax-b$ is strictly convex. Its two endpoint maxima must agree, giving $a=e-1$. Its unique interior minimum is at $c=\log a\in(0,1)$. Equating its magnitude to the endpoint error gives
$$
\boxed{p_1^*(x)=ax+\frac{1+a-a\log a}{2},\qquad
E_1=\frac{1-a+a\log a}{2},\qquad a=e-1.}
$$
The errors at $0,c,1$ are respectively $E_1,-E_1,E_1$. The <equioscillation theorem> therefore certifies this <best uniform approximation> directly.
For degree above one, subtracting a general <polynomial> does not preserve convexity of the error. The required $n+2$ alternating extreme points include additional unknown interior points, and their derivative and equal-height equations form a coupled nonlinear problem. The simple endpoint-and-single-minimum calculation therefore does not extend as a direct higher-degree recipe.
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