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Mean value theorem (MVT), it is also named as Lagrange's Mean Value Theorem.1) it is continuous on [a,b].

2) it is differentiable on (a,b).

Then, there exists a real number c $\epsilon$ (a,b) such that

$f'(c) = \frac{f(b)-f(a)}{b - a} $

Let f(x) be a function defined on [a,b] and let APB be a curve represented by y = f(x). Then coordinates of A and B are (a,f(a)) and (b,f(b)) respectively and P (c,f(c)) Suppose the chord AB makes ant angle $ \theta $ with the x-axis. Then we got a triangle as ARB and so we have ,

tan $ \theta = \frac{BR}{AR} $

$\Rightarrow tan \theta = \frac{f(b) -f(a)}{b - a }$

Slope of chord AB = slope of the tangent at P(c,f(c)).

Slope of tangent = tan $\theta $ = f '(c)

tan $\theta = \frac{f(b) -f(a)}{b - a }$

f '(c) = $\frac{f(b) -f(a)}{b - a }$

According to MVT,

f '(c) = $= \frac{f(b) -f(a)}{b - a }$

f '(c) = $= \frac{f(5) -f(3)}{5 - 3 }$ ------ (1)

Now,

f(x) = $x^{3} -18x^{2} + 99x -162$

f(5) = $5^{3} -18(5)^{2} + 99(5) -162$

= 125 - 450 + 495 -162

f(5) = 8

f(x) = $x^{3} -18x^{2} + 99x -162$

f(3) = $3^{3} -18(3)^{2} + 99(3) -162$

= 27 - 162 + 297 -162

f(3) = 0

So equation (1) becomes

f '(c) = $= \frac{8 -0 }{5 - 3 }$

f '(c) = $= \frac{8 }{2 }$

f '(c) = 4

Now, f(x) = $x^{3} -18x^{2} + 99x -162$

f '(x) = $3x^{2} -36x + 99$

f '(c) = $3c^{2} -36c + 99$

$3c^{2} -36c + 99$ = 4

$3c^{2} -36c + 95$ = 0

Now we will use a quadratic formula to fid the value of c

$x = -b \pm\frac{\sqrt{b^{2}-4ac}}{2a}$

$c = 36 \pm\frac{\sqrt{1296-1140}}{6}$

= $36 \pm\frac{12.49}{6}$

c = 8.8 , 4.8

Thus, c = 4.8 $\epsilon$ (3,5) such that f '(c) = $= \frac{f(5) -f(3)}{5 - 3 }$

Hence MVT is verified.

f(x) = 2- $3x^{2}$

f '(x) = -6x

f '(c) = -6c

f(x) = 2- $3x^{2}$

f(-1) = 2 - $3(-1)^{2}$ = -1

f(2) = 2 - $3(2)^{2}$ = -10

f '(c) = $\frac{f(2) -f(-1)}{2 -(-1) }$

-6c = $\frac{-10 -(-1)}{2 + 1 }$

-6c = $\frac{-9}{3 }$

c = $\frac{-3}{-6 }$

c = $\frac{1}{2 }$

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