Discussion
If \(f(x) = (2x + 3)e^{5x}\) , then \(f''(1) - 10f'(1)\) is equal to
Answer: E
The value is \(-125e^{5}\).
Differentiate with the product rule.
\[ \begin{aligned} f'(x) &= 2e^{5x} + 5(2x + 3)e^{5x} = (10x + 17)e^{5x} \\ f''(x) &= 10e^{5x} + 5(10x + 17)e^{5x} = (50x + 95)e^{5x} \\ f''(1) - 10f'(1) &= 145e^{5} - 270e^{5} = -125e^{5} \end{aligned} \]
The positive \(125e^{5}\) reverses the sign. Note that \(f'' - 10f' = -25(2x + 3)e^{5x}\) for every x, which at x = 1 gives the same \(-125e^{5}\).
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