Differential Calculus-2
Differential Calculus-2
Allen Star Batch
Grade 12
Question:
Let $f(x)$ be a function such that its derivative $f'(x)$ is continuous in $[a, b]$ and derivable in $(a, b)$. Consider a function $\phi(x) = f(b) - f(x) - (b-x)f'(x) - (b-x)^2A$. If Rolle's theorem is applicable to $\phi(x)$ on $[a, b]$ and for some $c \in (a, b), \phi'(c) = 0$ and $f(b) = f(a) + (b-a)f'(a) + xf''(c)(b-a)^2$ then $6A$ is equal to:
Step-by-Step Solution
Key Concept: Apply Rolle's theorem to φ(x) by ensuring φ(a) = φ(b), which yields the relationship between f(b), f(a), f'(a), and the constant A. Then use φ'(c) = 0 to extract f''(c) = 2A, and match coefficients with the Taylor expansion condition to solve for A.
From $\phi'(c) = 0$, we get $f''(c) = 2A$. Using the condition $\phi(a) = \phi(b)$, we expand and simplify: $f(b) - f(a) = (b-a)f'(a) + (b-a)^2 A$. Matching coefficients with the given form yields $\lambda = 1/2$.
Correct Answer: 3