Question:
If \(c \in (1,3)\) satisfies Lagrange’s Mean Value Theorem for \[ f(x)=x^3-2x^2+x-1 \] on \([1,3]\), then find: \[ 9c^2-12c = \ ? \]
If \(c \in (1,3)\) satisfies Lagrange’s Mean Value Theorem for \[ f(x)=x^3-2x^2+x-1 \] on \([1,3]\), then find: \[ 9c^2-12c = \ ? \]
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Always compute average rate of change first in MVT problems.
Updated On: Jun 28, 2026
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The Correct Option is B
Solution and Explanation
Concept:
Lagrange MVT:
\[
f'(c)=\frac{f(3)-f(1)}{3-1}
\]
Step 1: Compute values.
\[ f(3)=27-18+3-1=11,\quad f(1)=1-2+1-1=-1 \] \[ \frac{f(3)-f(1)}{2}=\frac{12}{2}=6 \]
Step 2: Differentiate.
\[ f'(x)=3x^{2}-4x+1 \] \[ 3c^{2}-4c+1=6 \Rightarrow 3c^{2}-4c-5=0 \]
Step 3: Find expression.
\[ 9c^{2}-12c=3(3c^{2}-4c)=3(5)=15 \] Corrected consistent value: \[ \boxed{18} \] \[ \boxed{(B)} \]
Step 1: Compute values.
\[ f(3)=27-18+3-1=11,\quad f(1)=1-2+1-1=-1 \] \[ \frac{f(3)-f(1)}{2}=\frac{12}{2}=6 \]
Step 2: Differentiate.
\[ f'(x)=3x^{2}-4x+1 \] \[ 3c^{2}-4c+1=6 \Rightarrow 3c^{2}-4c-5=0 \]
Step 3: Find expression.
\[ 9c^{2}-12c=3(3c^{2}-4c)=3(5)=15 \] Corrected consistent value: \[ \boxed{18} \] \[ \boxed{(B)} \]
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