Question

verify mean value theorem for the function f(x) = x^3-5x^2-3x in the interval [1,3]?​

Answer 1

The given function is  f(x)=x3−5x2−3x

The given function is  f(x)=x3−5x2−3xf being a polynomial function, so it is continuous in [1,3] and is differentiable in (1,3) whose derivative is 3x2−10x−3.

The given function is  f(x)=x3−5x2−3xf being a polynomial function, so it is continuous in [1,3] and is differentiable in (1,3) whose derivative is 3x2−10x−3.f (1) = 13 − 5⋅12 − 3⋅1 = −7,

f (3) = 33 − 5⋅32 − 3⋅3 = − 27

27∴b − af (b) − f (a) = 3 − 1f (3) − f(1) = 3 − 1 − 27 −(−7) = −10

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

−10⇒3c2 − 10c − 3 = −10

−10⇒3c2 − 10c + 7 = 0

0⇒3c2 − 3c − 7c + 7 = 0

0⇒3c(c−1)−7(c−1)=0

0⇒3c(c−1)−7(c−1)=0⇒(c−1)(3c−7)=0

0⇒3c(c−1)−7(c−1)=0⇒(c−1)(3c−7)=0⇒c = 0