If f(x) = x^3 - 6x^2 + 9x + 3 is a monotonica | vedclass

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(B) Given function: $f(x) = x^3 - 6x^2 + 9x + 3$. Find the derivative: $f'(x) = 3x^2 - 12x + 9$. For a function to be monotonically decreasing,the con

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$(1, 3)$

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(B) Given function: $f(x) = x^3 - 6x^2 + 9x + 3$. Find the derivative: $f'(x) = 3x^2 - 12x + 9$. For a function to be monotonically decreasing,the condition is $f'(x) So,$3x^2 - 12x + 9 Divide by $3$: $x^2 - 4x + 3 Factorize the quadratic expression: $(x - 3)(x - 1) Using the sign scheme for the inequality,the expression is negative between the roots $1$ and $3$. Therefore,$x \in (1, 3)$.

If $f(x) = x^3 - 6x^2 + 9x + 3$ is a monotonically decreasing function,then $x$ lies in:

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