Cauchy's Mean Value Theorem is defined as:
g′(c)f′(c)=g(b)−g(a)f(b)−f(a)
Here, given that
f(x)=x2 and g(x)=x and the interval is [a,b]
Then,
f′(x)=2xf′(c)=2c and g′(x)=1g′(c)=1
Therefore, as per Cauchy's Mean Value Theorem
g′(c)f′(c)=g(b)−g(a)f(b)−f(a)
12c=b−ab2−a2
12c=b−a(b−a)(b+a)
2c=a+b
Hence the Cauchy's Mean Value Theorem gives,
a+b=2c
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