Given the function $f(x)=-x^2+8x-17$ , how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [3,6] and find the c?

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Answer 1 · 2016-09-04T07:27:32

$c=9/2=4.5 in (3,6)$ .

The Mean Value Theorem :- If $f : [a,b] rarr RR$ is (1) continuous on

$[a,b]$ and (2) differentiable on $(a,b)$ , then $EE c in (a,b)$ such that,

$f'(c)=(f(b)-f(a))/(b-a)$ .

Here, $f : [3,6] rarr RR : f(x)=-x^2+8x-17$ .

$f$ is a Quadratic Poly. We know that Polys. are cont. &

differentiable on $RR$ , hence $f$ is cont. on $[3,6] sub RR,$ &, it is

differentiable on $(3,6)$ .

$:. f'(c)=(f(6)-f(3))/(6-3)," for some c in "[3,6].................(star)$ .

Now, $f(x)=-x^2+8x-17 :. f'(x)=-2x+8$

$:. f'(c)=-2c+8....................................................(1)$

Also,

$(f(6)-f(3))/(6-3)={(-36+48-17)-(-9+24-17)}/3$ , i.e.,

$(f(6)-f(3))/(6-3)=-3/3=-1...............................(2)$ .

Hence, by $(1),(2), and, (star)$ , we have,

$-2c+8=-1 rArr -2c=-9 rArr c=9/2=4.5 in (3,6)$ .

Enjoy maths.!

Given the function f(x)=-x^2+8x-17f(x)=-x^2+8x-17, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [3,6] and find the c?