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(D) Given the function $f(x) = x^3 - 12x + \lambda$. To find the points of local maxima and minima,we calculate the first derivative: $f'(x) = 3x^2 -

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$21$

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(D) Given the function $f(x) = x^3 - 12x + \lambda$. To find the points of local maxima and minima,we calculate the first derivative: $f'(x) = 3x^2 - 12$. Setting $f'(x) = 0$,we get $3(x^2 - 4) = 0$,which implies $x = 2$ and $x = -2$. Using the second derivative test,$f''(x) = 6x$. At $x = -2$,$f''(-2) = -12 At $x = 2$,$f''(2) = 12 > 0$,so the function has a local minimum at $x = 2$. Since the existence of local maxima and minima for a cubic polynomial depends only on the derivative $f'(x)$,which is independent of the constant $\lambda$,the function $f(x)$ has a point of local maxima for all values of $\lambda$. Given $\lambda \in [0, 20]$,the integral values of $\lambda$ are ${0, 1, 2, \dots, 20}$. The total number of such 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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