Q. 95.0( 7 Votes )

If f

Answer :

Given: f:[-5, 5] R is a differentiable function and f(x) does not vanish anywhere


To prove: f(-5) f(5)


Since f(x) is differentiable function in [-5, 5]


We know, every differentiable function is continuous too


f(x) is continuous and differentiable function in [-5, 5]


Lagrange’s mean value theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there is at least one point x=c on this interval, such that


f(b)−f(a)=f′(c)(b−a)



This theorem is also known as First Mean Value Theorem.


As, f’(x) does not vanish anywhere


f’(x) ≠ 0 for any value of x


Thus f’(c) ≠ 0





f(5) – f(-5) ≠ 0 × 10


f(5) – f(-5) ≠ 0


f(5) f(-5)


Hence Proved


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