A function f is defined by f : x ↦ (2x − 3)3 − 8, for 2 ≤ x ≤ 4.
Find an expression, in terms of x, for f'(x) and show that f is an increasing function.
Solution
\(To find f'(x), we differentiate f(x) = (2x − 3)3 − 8.\)
\(Using the chain rule, let u = 2x − 3, then f(x) = u3 − 8.\)
The derivative of u3 is 3u2, and the derivative of u with respect to x is 2.
\(Thus, f'(x) = 3(2x − 3)2 × 2 = 6(2x − 3)2.\)
Since (2x − 3)2 is always positive for all x, f'(x) is positive for all x in the interval [2, 4].
Therefore, f is an increasing function.
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