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(B) function $f(x)$ is invertible if and only if it is strictly monotonic (either strictly increasing or strictly decreasing) on its domain.For a cubi

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$1$

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(B) function $f(x)$ is invertible if and only if it is strictly monotonic (either strictly increasing or strictly decreasing) on its domain.For a cubic polynomial $f(x)$,this occurs when $f'(x) \ge 0$ or $f'(x) \le 0$ for all $x \in R$.First,find the derivative: $f'(x) = 6x^2 - 6(a + 2)x + 12a$.Factor the derivative: $f'(x) = 6(x^2 - (a + 2)x + 2a) = 6(x - a)(x - 2)$.For $f(x)$ to be monotonic,$f'(x)$ must not change sign. However,since $f'(x)$ is a quadratic with two roots $a$ and $2$,it changes sign unless the roots are equal.Thus,we must have $a = 2$.If $a = 2$,$f'(x) = 6(x - 2)^2 \ge 0$ for all $x$,so the function is strictly increasing and thus invertible.Since $a = 2$ is the only integral value in the interval $[-4, 6]$ that satisfies the condition,the number of such values is $1$.

The number of integral values of $a$ for which the function $f: R \to R, f(x) = 2x^3 - 3(a + 2)x^2 + 12ax - 7$ where $a \in [-4, 6]$ is invertible,is

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