Multiple choice

If a is the double root of the equation x3 - 5x2 + 3x + 9 = 0, a is also the root of

  1. f ′(x) = 0

  2. f ′ (x) + f ′′(x) = 0

  3. f (x) = 0

  4. f (x) + f ′′ (x) = 0
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A Correct answer
Explanation

If 'a' is a double root of f(x) = 0, then f(a) = 0 and f'(a) = 0. Therefore, 'a' must be a root of f'(x) = 0.

AI explanation

If a is a double root of the polynomial equation f(x) = 0, then (x - a)2 is a factor of f(x). Using the product rule, this means f(x) can be written as (x - a)2 multiplied by another linear factor. When taking the first derivative, f ′(x) will still contain (x - a) as a factor, making a root of the derivative as well. Therefore, a is also a root of the equation f ′(x) = 0.

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