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The number of distinct real roots of x^4...

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

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The correct Answer is:
2
Let ` f(x) = x^(4) - 4x ^(3) + 12 x^(2) + x - 1 `
` therefore f'(x) = 4x ^(3) - 12 x^(2) + 24 x + 1`
and ` f''(x) = 12 x^(2) - 24x + 24 gt 0 ` for all real x.
So, graph of f'(x) intersects x - axis only once .
Hene, f(x) has only one turning point .
Also `f(0) = - 1` .
so, graph of f(x) cuts x-axis at two points
Hence , f(x) = 0 has two real roots .
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