Let $(2,3)$ be the largest open interval in which the function $f(x)=2\log_e(x-2)-x^2+ax+1$ is strictly increasing and $(b,c)$ be the largest open interval, in which the function $g(x)=(x-1)^3(x+2-a)^2$ is strictly decreasing. Then $100(a+b-c)$ is equal to :

Question

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] $f'(x)=\frac2{x-2}-2x+a$ must vanish at $x=3$: $2-6+a=0\Rightarrow a=4$, and then $f'=\frac{-2(x-1)(x-3)}{x-2}>0$ on $(2,3)$. For $g(x)=(x-1)^3(x-2)^2$: $g'=(x-1)^2(x-2)(5x-8)

Q. Let $(2,3)$ be the largest open interval in which the function $f(x)=2\log_e(x-2)-x^2+ax+1$ is strictly increasing and $(b,c)$ be the largest open interval, in which the function $g(x)=(x-1)^3(x+2-a)^2$ is strictly decreasing. Then $100(a+b-c)$ is equal to :

Explanation

Reviewed by Dr. Abubacker

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