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Question
Verify MVT if f(x)=x3−5x2−3x iin the interval [a, b], where a = 1 and b = 3. Find all cϵ(1, 3) for which f'(c) = 0.
Verify MVT if f(x)=x3−5x2−3x iin the interval [a, b], where a = 1 and b = 3. Find all cϵ(1, 3) for which f'(c) = 0.
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Solution
Given, f(x)=x3−5x2−3x, xϵ(1,3), which is a polynomial function. Since, a polynomial function is continuous and derivable at all x ϵR, therefore
(i) f(x) is continuous on [1, 3]. (ii) f(x) is derivable on (1, 3).
∴ Condition of Lagrange's MVT are satisfied on [1, 3].
Hence, there exists atleast one real cϵ(1, 3).
Such that f'(c)=f(3)−f(1)3−1
⇒ 3c2−10c−3=(33−5×32−3×3)−(1−5−3)3−1 (∴ f′(c)=−f(b)−f(a)b−a) (∵ f′(x)=ddx(x3−5x2−3x)=3x2−10x−3)3c2−10c−3=−10⇒ 3c2−10c+7=0⇒ c=10±√100−846=10±46=1, 73Note that 73ϵ(1, 3)
Given, f(x)=x3−5x2−3x, xϵ(1,3), which is a polynomial function. Since, a polynomial function is continuous and derivable at all x ϵR, therefore
(i) f(x) is continuous on [1, 3]. (ii) f(x) is derivable on (1, 3).
∴ Condition of Lagrange's MVT are satisfied on [1, 3].
Hence, there exists atleast one real cϵ(1, 3).
Such that f'(c)=f(3)−f(1)3−1
⇒ 3c2−10c−3=(33−5×32−3×3)−(1−5−3)3−1 (∴ f′(c)=−f(b)−f(a)b−a) (∵ f′(x)=ddx(x3−5x2−3x)=3x2−10x−3)3c2−10c−3=−10⇒ 3c2−10c+7=0⇒ c=10±√100−846=10±46=1, 73Note that 73ϵ(1, 3)
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in the interval [a, b], where a = 1 and b
= 3. Find all
for
which
