Finding Smallest Domain Value for Inverse Function
For a function $f(x)$ to have an inverse on a domain $[a, \infty)$, it must be strictly monotonic (either strictly increasing or strictly decreasing) throughout that domain.
Analyzing Function Monotonicity
We are given the function $f(x) = 2x^3 - 3x^2 + 6$. To determine its monotonicity, we first find its derivative, $f'(x)$.
$ f'(x) = \frac{d}{dx}(2x^3 - 3x^2 + 6) = 6x^2 - 6x $Now, we find the critical points by setting the derivative to zero:
$ 6x^2 - 6x = 0 $ $ 6x(x - 1) = 0 $This gives us critical points at $x = 0$ and $x = 1$. We analyze the sign of $f'(x)$ in the intervals defined by these points:
- For $x < 0$, $f'(x) > 0$ (e.g., $f'(-1) = 6(-1)(-2) = 12 > 0$). The function $f(x)$ is increasing.
- For $0 < x < 1$, $f'(x) < 0$ (e.g., $f'(0.5) = 6(0.5)(-0.5) = -1.5 < 0$). The function $f(x)$ is decreasing.
- For $x > 1$, $f'(x) > 0$ (e.g., $f'(2) = 6(2)(1) = 12 > 0$). The function $f(x)$ is increasing.
Determining the Domain for Invertibility
The function $f(x)$ must be strictly monotonic on the domain $[a, \infty)$ for it to have an inverse.
- If $a=0$, the domain is $[0, \infty)$. $f(x)$ decreases on $(0, 1)$ and increases on $(1, \infty)$, so it's not monotonic.
- If $a=-1$, the domain is $[-1, \infty)$. $f(x)$ increases on $[-1, 0)$ and decreases on $(0, 1)$, so it's not monotonic.
- For the function to be strictly increasing on $[a, \infty)$, we need $f'(x) > 0$ for all $x \in [a, \infty)$. Since $f'(x) = 0$ at $x=1$, we must start the domain after $x=1$ to ensure strict increase. Thus, we require $a > 1$.
The smallest value of '$a$' from the given options that satisfies $a > 1$ is $a=2$.
Calculating Range for $a=2$
If $a=2$, the domain is $[2, \infty)$. On this interval, $f'(x) > 0$, so the function is strictly increasing and has an inverse.
We need to find the range $B = [f(a), \infty)$.
Calculate $f(a)$ for $a=2$:
$ f(2) = 2(2)^3 - 3(2)^2 + 6 $ $ f(2) = 2(8) - 3(4) + 6 $ $ f(2) = 16 - 12 + 6 $ $ f(2) = 10 $Therefore, the range is $B = [10, \infty)$.
Conclusion
The smallest value of '$a$' among the options for which $f(x)$ is strictly monotonic on $[a, \infty)$ and thus has an inverse is $a=2$. The corresponding range is $B = [10, \infty)$.
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