Question

If is a differentiable function and if does not vanish anywhere, then prove that .

Answer 1

Let, f:[ −5,5 ]→Rand it is a differentiable function.

The conditions for the differentiable function are,

(a) It is continuous on the close interval [ −5,5 ].

(b) It is differentiable on the open interval ( −5,5 ).

According to Mean Value Theorem, there exists c∈( −5,5 )such that,

f ′ ( c )= f( 5 )−f( −5 ) { 5−( −5 ) } 10 f ′ ( c )=f( 5 )−f( −5 )

According to question f ′ ( x )does not vanish anywhere.

f ′ ( c )≠0 10 f ′ ( c )≠0 f( 5 )−f( −5 )≠0 f( 5 )≠f( −5 )

Hence, the value of function f( 5 )is not equal to the value of function f( −5 ).