Question
Find all quadratic polynomials that are orthogonal to the function $e^x$ with respect to the $\mathrm{L}^2$ inner product on the interval $[0,1]$.
Step 1
\] We need to find all quadratic polynomials $p(x) = ax^2 + bx + c$ that are orthogonal to $e^x$ under this inner product. This means we need: \[ \langle p, e^x \rangle = \int_0^1 (ax^2 + bx + c) e^x \, dx = 0. \] Show more…
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