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Solve the following.
Given 2f(x) = log|x|^a - bx² + x, if f(x) has extreme values at x = -1 and x = 2, find the value of…
Given $2f(x) = \log|x|^a - bx^2 + x$, if $f(x)$ has extreme values at $x = -1$ and $x = 2$, find the value of $a$ and $b$.
Answer
Answer
Official answer key$f(x) = \frac{a}{2}\log|x| - \frac{1}{2}bx^2 + \frac{1}{2}x$
$f'(x) = \frac{a}{2x} - bx + \frac{1}{2}$
At $x = -1$, $-a + 2b = -1$
At $x = 2$, $a - 8b = -2$
$a = 2, b = \frac{1}{2}$
Final answer: $a = 2, b = \frac{1}{2}$