Hello, Runty!
I have an intuitive (graphic) proof . . .
Suppose is not equal to 0 for all
. . Then is not the axis.
There are three cases:
. . (1) is above the xaxis on the interval
. . (2) is below the xaxis on the inverval
. . (3) crosses the xaxis on the interval
For case (1), we have this graph: Code:

 *
 *:
 *:::
 *:::::
 *:::::::
 :::::::
    +    +  
 a b
represents the area under the curve,
. . . which is evidently not zero.
For case (2), we have this graph: Code:

 a b
  +  +    +  
 :::::::
 *:::::::
 *:::::
 *:::
 *:
 *
represents the area above the curve,
. . . which also is not zero.
For case (3), we have this graph for : Code:

 *
 *
 *
 *
  +  +    *   +   
 a * b
 *
 *
The the graph of looks like this: Code:

 *
 :* *
 :::* *:
 :::::* *:::
    +    *    +   
 a b

represents the area under the curve,
. . . which again is not zero.
[This may not be considered a a satisfactory proof.]