Test whether the following function f(x) = 2 – 3x + 3x^2 – x^3, x… - Vidyadip

Question

Test whether the following function $f(x) = 2 – 3x + 3x^2 – x^3, x \in R$ is increasing or decreasing

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Answer

$f(x) = 2 – 3x + 3x^2 – x^3$
$\therefore f′(x) = – 3 + 6x – 3x^2$
$= –3(x^2 – 2x + 1)$
$= –3(x – 1)^2$
$(x – 1)^2$ is always positive for $x ≠ 1$ and $– 3 < 0$.
$\therefore –3(x – 1)^2$ is always negative for $x ≠ 1$.
$\therefore f′(x) \leq 0$ for all $x \in R.$
Hence, f(x) is a decreasing function for all $x \in R.$

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