It is given that \(f(x) = (2x - 5)^3 + x\), for \(x \in \mathbb{R}\). Show that \(f\) is an increasing function.
Solution
To determine if \(f(x) = (2x - 5)^3 + x\) is increasing, we need to find its derivative \(f'(x)\).
Using the chain rule, the derivative of \((2x - 5)^3\) is \(3(2x - 5)^2 \times 2\).
Thus, \(f'(x) = 3(2x - 5)^2 \times 2 + 1 = 6(2x - 5)^2 + 1\).
We can also express this as \(f'(x) = 24\left(\frac{x - 5}{2}\right)^2 + 1\).
Since \((2x - 5)^2 \geq 0\) for all \(x\), it follows that \(6(2x - 5)^2 + 1 > 0\) for all \(x\).
Therefore, \(f'(x) > 0\) for all \(x \in \mathbb{R}\), which means \(f\) is an increasing function.
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