Question

The e of the mean value theorem
for the funation f(x)= x(2-x), where
a=0 and b=1.5 is​

Answer 1

Step-by-step explanation:

The given function is f(x)=x2−4x−3

f being a polynomial function, is continuous in [1,4] and is differentiable in (1,4) and derivative is f′(x)=2x−4.

Now f(1)=12−4⋅1−3=−6,f(4)=42−4⋅4−3=−3

∴b−af(b)−f(a)=4−1f(4)−f(1)=3−3−(−6)=33=1

Mean Value Theorem states that there is a point c ∈(1,4) such that f′(c)=b−af(b)−f(a)=1

⇒2c−4=1

⇒c=25∈(1,4)

Hence, Mean Value Theorem is verified for the given function.

Given f(x)−logxf′(x)=x1

using Langrage's mean value theorm

f′(c)=b−af(b)−f(a)⇒c1=e−1loge−log1⇒c1=e−11⇒c=e−1