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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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The correct Answer is:
9
Plan : If ` f(x)` is least degree polynominal having local maximum and local minimum at ` alpha and beta `
Then, ` f' (x) = lamda (x - alpha) ( x - beta )`
Here, ` p ' (x) = lamda (x - 1 ) (x - 3) = lamda (x ^(2) - 4x + 3)`
On integrating both sides between 1 to 3, we get
`int_(1)^(3) p '(x) dx = int_(1)^(3) lamda (x^(2) - 4x + 3) dx `
`rArr ( p ( x ))_1^(3) = lamda ((x ^(3))/( 3) - 2x ^(2) + 3x )_(1)^(3) `
`rArr p (3) - p (1) = lamda ( ( 9 - 18 + 9) - ((1)/(3) - 2 + 3))`
`rArr 2 - 6 = lamda{(-4)/(3)}`
`rArr " " lamda = 3 `
` rArr p ' (x) = 3 (x - 1 ) (x - 3)`
`therefore p'(0) = 9`
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