A Tangent-Determined Function

★ 7/10functions Extrema calculus Monotonicity

Problem

Let $f(x) = a\ln x - \dfrac{x - b}{x + 1}$ , and suppose the tangent line to $y = f(x)$ at $(1, f(1))$ is $x + 4y - 1 = 0$ .

1. Find $a$ and $b$ . 2. Find the intervals of monotonicity of $f(x)$ . 3. Find the number of zeros of

g(x) = (x^2 - 1)\ln x - 4(x - 1)^2.

Answer

Solution

Difficulty	7/10
Topics	functions, Extrema, calculus, Monotonicity

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