If the function $f(x) = ax^3-bx^2-8x-4$ satisfies Roll's theorem in $[1,3]$, if $f^'(2) = 0$ then $a-b$ is equal to...

Question

If $f(x)$ and $g(x)$ are differentiable functions for $0 \leq x \leq 1$ such that, $f(1)-f(0) = k(g(1)-g(0))$, $k \neq 0$, and there exists a 'c' satisfying $0<c<1$. Then, the value of $\frac{f'(c)}{g'(c)}$ is equal to

Answer 1

Step 1: Alternative route:
Eliminate $4b$ between $13a-4b=8$ and $12a-4b=8$ (the second is $f'(2)=0$ rewritten).

Step 2: Subtract:
$(13a-4b)-(12a-4b)=8-8$ gives $a=0$.

Step 3: Then:
$-4b=8$, so $b=-2$.

Step 4: Answer:
$a-b=2$. Option (A).

Final Answer:
Subtracting the two linear equations gives a = 0. \[ \boxed{A} \]

If the function \(f(x) = ax^3-bx^2-8x-4\) satisfies Roll's theorem in \([1,3]\), if \(f^'(2) = 0\) then \(a-b\) is equal to...

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The Correct Option is A

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