Answer to Question #257102 in Calculus for Nazmul

Question #257102

Verify mean value theorem, if f(x) =x2-4x-3 in the interval [a,b] where a= 1 and b= 4 x


1
Expert's answer
2021-10-27T12:26:43-0400

f(x)=x24x3f(x)=x^2-4x-3 is continuous on [1,4][1,4] as polynomial.


f(x)=x24x3f(x)=x^2-4x-3 is differentiable on (1,4)(1,4) as polynomial.


Therefore, by the Mean Value Theorem, there is a number cc in (1,4)(1, 4) such that


f(4)f(1)=f(c)(43)f(4)-f(1)=f'(c)(4-3)

f(4)=424(4)3=3f(4)=4^2-4(4)-3=-3

f(1)=124(1)3=6f(1)=1^2-4(1)-3=-6

f(x)=2x4f'(x)=2x-4

f(c)=2c4f'(c)=2c-4

Substitute


3(6)=(2c4)(41)-3-(-6)=(2c-4)(4-1)

2c4=12c-4=1

c=52c=\dfrac{5}{2}

1<52<4=>c(1,4)1<\dfrac{5}{2}<4=>c\in (1, 4)

Hence the hypotheses of the Mean Value Theorem are satisfied.



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