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Let p(x) be a real polynomial of least d...

Let `p(x)` be a real polynomial of least degree which has a local maximum at `x=1` and a local minimum at `x=3.` If `p(1)=6a n dp(3)=2,` then `p^(prime)(0)` is_____

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To solve the problem, we need to find the polynomial \( p(x) \) that has a local maximum at \( x = 1 \) and a local minimum at \( x = 3 \). We are given that \( p(1) = 6 \) and \( p(3) = 2 \). We will also find \( p'(0) \). ### Step 1: Determine the form of \( p'(x) \) Since \( p(x) \) has a local maximum at \( x = 1 \) and a local minimum at \( x = 3 \), the derivative \( p'(x) \) must be zero at these points. Therefore, we can express \( p'(x) \) as: \[ p'(x) = k(x - 1)(x - 3) ...
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