The number of distinct real roots of x^4-4x^3 | vedclass

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(A) Let $f(x) = x^4 - 4x^3 + 12x^2 + x - 1$.First,find the derivative $f'(x) = 4x^3 - 12x^2 + 24x + 1$.Now,find the second derivative $f''(x) = 12x^2

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$2$

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(A) Let $f(x) = x^4 - 4x^3 + 12x^2 + x - 1$.First,find the derivative $f'(x) = 4x^3 - 12x^2 + 24x + 1$.Now,find the second derivative $f''(x) = 12x^2 - 24x + 24 = 12(x^2 - 2x + 2)$.The discriminant of the quadratic $x^2 - 2x + 2$ is $D = (-2)^2 - 4(1)(2) = 4 - 8 = -4 Since $D Thus,$f'(x) = 0$ has exactly one real root.Since $f'(x)$ has only one real root,$f(x)$ can have at most two real roots.Evaluating $f(x)$ at some points: $f(0) = -1$ and $f(1) = 9$.Since the sign changes between $x=0$ and $x=1$,there is at least one root in $(0, 1)$.Also,$f(-1) = 1 + 4 + 12 - 1 - 1 = 15 > 0$,so there is at least one root in $(-1, 0)$.Therefore,the equation has exactly $2$ distinct real roots.

The number of distinct real roots of $x^4-4x^3+12x^2+x-1=0$ is

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