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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

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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 $f'(x) = 0$ gives $3(x^2 - 4) = 0$,which implies $x = 2$ or $x = -2$.Now,we check the second derivative:$f''(x) = 6x$.At $x = -2$,$f''(-2) = 6(-2) = -12 At $x = 2$,$f''(2) = 6(2) = 12 > 0$,which indicates a local minimum at $x = 2$.Since the existence of local maxima and minima for a cubic polynomial $f(x) = ax^3 + bx^2 + cx + d$ depends only on the derivative $f'(x) = 3ax^2 + 2bx + c$,and here $f'(x) = 3x^2 - 12$ is independent of $\lambda$,the function $f(x)$ will have 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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