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Solve the following.
Given f(x) = 2log(x-2) - x² + 4x + 1 and f'(x) = (-k(x-p)(x-q))/(x-k). Find k + p + q. The…
Given $f(x) = 2\log(x-2) - x^2 + 4x + 1$ and $f'(x) = \frac{-k(x-p)(x-q)}{(x-k)}$.
(a)[1]
Find $k + p + q$.
(b)[1.5]
The Statement “$f(x)$ is strictly increasing in $(-\infty, 1] \cup (2, 3]$” is false. Justify.
(c)[1.5]
Hence, find the interval(s) in which the function is strictly increasing.
Answer
Answer (a)
Official answer key$k + p + q = 6$.
Final answer: 6
Answer (b)
Official answer key$f'(x) \geq 0$, $\forall x \in (-\infty, 1] \cup (2, 3]$. But domain of $f(x)$ is $x \in (2, \infty)$.
$\therefore$ The Statement “$f(x)$ is strictly increasing in $(-\infty, 1] \cup (2, 3]$” is false.
Answer (c)
Official answer key$f(x)$ is strictly increasing in $(2, 3]$.
Final answer: $(2, 3]$