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(D) Let $f(x) = ax^3 + bx^2 + cx + d$ be a polynomial of degree $3$,where $a 
eq 0$.Then $f'(x) = 3ax^2 + 2bx + c$ is a quadratic polynomial.Stationa

Vedclass · Accepted Answer

$f$ has either $0$ or $2$ stationary points.

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(D) Let $f(x) = ax^3 + bx^2 + cx + d$ be a polynomial of degree $3$,where $a 
eq 0$.Then $f'(x) = 3ax^2 + 2bx + c$ is a quadratic polynomial.Stationary points are the roots of $f'(x) = 0$.$A$ quadratic equation can have $0, 1,$ or $2$ real roots.If $f'(x) = 0$ has exactly one root $\alpha$,then $\alpha$ must be a repeated root,so $f'(x) = 3a(x - \alpha)^2$.In this case,$f''(x) = 6a(x - \alpha)$,which implies $f''(\alpha) = 0$.However,the problem states that $f''(x) 
eq 0$ at any stationary point.Therefore,$f'(x) = 0$ cannot have a repeated root.Thus,$f'(x) = 0$ must have either $0$ distinct real roots or $2$ distinct real roots.Hence,$f$ has either $0$ or $2$ stationary points.

Suppose that $f$ is a polynomial of degree $3$ and that $f''(x) \neq 0$ at any of the stationary points. Then

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