Why Is the Slope of a Beam Zero at Its Midpoint Despite Maximum Deflection?

[influx]

Can someone explain why the slope dy/dx = 0 at x = L/2? L/2 is the midpoint and there would be a deflection here so surely the slope of the deflection curve shouldn't be 0? I'm finding it hard to visualise this.

EDIT: I think I understand the above. The slope of the deflection curve at x = L/2 will be 0 as the slope at this point is a flat line of constant value (due to the symmetry of the diagram). This is correct yes?

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Is the part circled in red a mistake? There shouldn't be a negative sign after integrating 0 = EI(d^4y/dx^4)? As in it should be just A = EI(d^3y/dx^3) rather than A = -EI(d^3y/dx^3)?

 

[SteamKing]

Can someone explain why the slope dy/dx = 0 at x = L/2? L/2 is the midpoint and there would be a deflection here so surely the slope of the deflection curve shouldn't be 0? I'm finding it hard to visualise this.

EDIT: I think I understand the above. The slope of the deflection curve at x = L/2 will be 0 as the slope at this point is a flat line of constant value (due to the symmetry of the diagram). This is correct yes?

The value of the slope of the beam is independent of the value of the deflection at the same location.

For the simply supported beam, the deflection at x = L/2 will be a maximum, and since the slope curve is the first derivative of the deflection curve, what value will the derivative of the deflection curve have where the deflection is a maximum? This is a basic property of derivatives from intro calculus.