Question

Verify Mean Value Theorem, if f (x) = x³-5x²-3x in the interval [a, b], where a =1 

c ∈ (1,3) for which  f ' (c)=0    b= 3. find all c 

Answer 1

The given function f is f (x) = x³ -5x²-3x
f, being a polynomial function, is continuous in [1, 3] and is differentiable in (1, 3)

whose derivative is 3x² − 10x − 3.

Mean Value Theorem states that there exist a point c∈ (1, 3) such that f' (c) =-10

Hence, Mean Value Theorem is verified for the given function and  c= 7/3 ∈ (1,3) is the only point for which f ' (c) =0