Question

Let \(A=[a,\infty)\) denotes the domain, then \(f:[a,\infty)\rightarrow B\) which is defined by \(f(x)=2x^{3}-3x^{2}+6\) will have an inverse for the smallest real value of \(a\) if:

• A. $a=0, \ B=[6,\infty)$
• B. $a=2, \ B=[10,\infty)$
• C. $a=1, \ B=[5,\infty)$
• D. $a=-1, \ B=[5,\infty)$

Hint:

A function can only have an inverse on intervals where its derivative does not change sign. On a graph, the critical points $x=0$ and $x=1$ are the local peaks and valleys where the curve turns around. To keep the function invertible all the way to infinity, we must start our domain at or after the final local valley at $x=1$.

Answer 1

The Correct Option is C

Concept: For a continuous function to have a valid inverse, it must be bijective (both one-to-one and onto). A continuous real function is one-to-one on an interval if and only if it is strictly monotonic—meaning it either increases across the entire interval or decreases across the entire interval without changing direction. Step 1: Find the derivative to locate the turning points.
Let us find the first derivative of the given cubic polynomial function: $$f(x) = 2x^3 - 3x^2 + 6 \quad \Rightarrow \quad f'(x) = 6x^2 - 6x$$ Factor the derivative expression to find the critical stationary points: $$6x(x - 1) = 0 \quad \Rightarrow \quad x = 0 \quad \text{and} \quad x = 1$$

Step 2: Analyze the intervals of increase and decrease.
Let us check the sign of $f'(x)$ across different regions of the number line using the intervals between our critical points:
• For $x \in (-\infty, 0)$: $f'(x) > 0 \implies$ Strictly Increasing
• For $x \in (0, 1)$: $f'(x) < 0 \implies$ Strictly Decreasing
• For $x \in (1, \infty)$: $f'(x) > 0 \implies$ Strictly Increasing

Step 3: Determine the smallest boundary value $a$.
The problem states that the domain is bounded on the left, forming the interval $[a, \infty)$. For the function to have an inverse, it must be strictly monotonic throughout this entire region, meaning the interval cannot contain any turning points. The function increases strictly for all values from $x = 1$ to infinity. Therefore, the smallest possible starting coordinate that ensures the function never changes direction is: $$a = 1$$

Step 4: Calculate the matching onto codomain set $B$.
Since our domain interval is $[1, \infty)$ and the function is strictly increasing across this region, the absolute minimum value occurs exactly at the left boundary $x = 1$: $$f(1) = 2(1)^3 - 3(1)^2 + 6 = 2 - 3 + 6 = 5$$ As $x$ grows towards infinity, the cubic curve grows towards infinity ($f(x) \rightarrow \infty$). Therefore, the output range set is: $$B = [5, \infty)$$ This matches option (C) perfectly.