Given lambda in [0, 20],find the number of in | vedclass

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(D) For a cubic function $f(x) = ax^3 + bx^2 + cx + d$ to have a point of local maxima and a point of local minima,its derivative $f'(x)$ must have tw

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$21$

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(D) For a cubic function $f(x) = ax^3 + bx^2 + cx + d$ to have a point of local maxima and a point of local minima,its derivative $f'(x)$ must have two distinct real roots.Given $f(x) = x^3 - 12x + \lambda$.Then $f'(x) = 3x^2 - 12$.Setting $f'(x) = 0$,we get $3x^2 = 12$,which implies $x^2 = 4$,so $x = \pm 2$.Since the derivative $f'(x) = 3(x-2)(x+2)$ changes sign at $x = -2$ and $x = 2$,the function $f(x)$ always has a local maximum at $x = -2$ and a local minimum at $x = 2$,regardless of the value of $\lambda$.Therefore,the function $f(x)$ has a point of local maxima for all real values of $\lambda$.Given $\lambda \in [0, 20]$,the integral values of $\lambda$ are ${0, 1, 2, \dots, 20}$.The number of such integral values is $20 - 0 + 1 = 21$.

Given $\lambda \in [0, 20]$,find the number of integral values of $\lambda$ for which the function $f(x) = x^3 - 12x + \lambda$ has a point of local maxima.

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