Question

`f(x)=-1/2x^2+6x-13`. The line `y=1/2x+c` is orthogonal to the graph of `f`. Then what is `c`?

Answer 1

`c = -1`

Explanation:

Find a point on the line. We'll find the point of intersection with the graph of `f`.

`f'(x) = -x+6`

The slope of the normal line is `1/2`, so the slope of the tangent line is `-2` and `f'(x) = -2` at `x = 8`

`f(8) = -4*8+6*8-13 = 2*8-13 = 3`

So the line `y = 1/2x+c` contains the point `(8,3)`.

So `3 = 1/2(8)+c` and `c = -1`