Let f(x) = (x-4)(x-5)(x-6)(x-7),then - | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given $f(x) = (x-4)(x-5)(x-6)(x-7)$.Since $f(x)$ is a polynomial of degree $4$,$f'(x)$ is a polynomial of degree $3$. Thus,$f'(x) = 0$ has $3$ roo

Vedclass · Accepted Answer

Three roots of $f'(x) = 0$ lie in $(4, 5) \cup (5, 6) \cup (6, 7)$

Vedclass · Answer

(B) Given $f(x) = (x-4)(x-5)(x-6)(x-7)$.Since $f(x)$ is a polynomial of degree $4$,$f'(x)$ is a polynomial of degree $3$. Thus,$f'(x) = 0$ has $3$ roots.According to Rolle's Theorem,if $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$ with $f(a) = f(b)$,then there exists at least one $c \in (a, b)$ such that $f'(c) = 0$.Here,$f(4) = f(5) = 0$,so there exists a root $c_1 \in (4, 5)$ such that $f'(c_1) = 0$.$f(5) = f(6) = 0$,so there exists a root $c_2 \in (5, 6)$ such that $f'(c_2) = 0$.$f(6) = f(7) = 0$,so there exists a root $c_3 \in (6, 7)$ such that $f'(c_3) = 0$.Thus,the three roots of $f'(x) = 0$ lie in the intervals $(4, 5)$,$(5, 6)$,and $(6, 7)$ respectively.Therefore,the three roots lie in $(4, 5) \cup (5, 6) \cup (6, 7)$.

Let $f(x) = (x-4)(x-5)(x-6)(x-7)$,then -

Similar Questions

Let the function $f:[-7,0] \rightarrow R$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f'(x) \leq 2$ for all $x \in (-7,0)$,then for all such functions $f$,$f(-1)+f(0)$ lies in the interval:

The constant $c$ of Rolle's theorem for the function $f(x)=(x-1)^3(x-2)^5$ in the interval $[1, 2]$ is:

Let $f: R \rightarrow R$ be a differentiable function such that $f(a)=0=f(b)$ and $f^{\prime}(a) f^{\prime}(b) > 0$ for some $a < b$. Then,the minimum number of roots of $f^{\prime}(x)=0$ in the interval $(a, b)$ is

Consider the following functions:
$I) f(x) = \begin{cases} \frac{1}{2}-x, & x < \frac{1}{2} \\ (\frac{1}{2}-x)^2, & x \geq \frac{1}{2} \end{cases}$
$II) f(x) = |3x-1|$
$III) f(x) = x|x|$
$IV) f(x) = |x|$
Then on $[0, 1]$,Lagrange's Mean Value Theorem $(LMVT)$ is applicable to which of the functions?

If $f(x)=(2x-1)(3x+2)(4x-3)$ is a real-valued function defined on $[\frac{1}{2}, \frac{3}{4}]$, then the value$(s)$ of '$c$' as defined in the statement of Rolle's theorem is/are:

Vedclass Test Series

Exam Paper Generator

Online Exam Module