Let f: mathbb{R} rightarrow mathbb{R} be a tw | vedclass

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

Question

(B) Given that $f(0)=0$ and $f(1)=0$. By Rolle's Theorem,there exists at least one $c_1 \in (0, 1)$ such that $f^{\prime}(c_1)=0$. We are also given $

Vedclass · Accepted Answer

$f^{\prime \prime}(c)=0$ for some $c \in (0, 1)$

Vedclass · Answer

(B) Given that $f(0)=0$ and $f(1)=0$. By Rolle's Theorem,there exists at least one $c_1 \in (0, 1)$ such that $f^{\prime}(c_1)=0$. We are also given $f^{\prime}(0)=0$. Now,consider the function $f^{\prime}(x)$ on the interval $[0, c_1]$. Since $f$ is twice continuously differentiable,$f^{\prime}$ is continuous on $[0, c_1]$ and differentiable on $(0, c_1)$. We have $f^{\prime}(0)=0$ and $f^{\prime}(c_1)=0$. By applying Rolle's Theorem to $f^{\prime}(x)$ on the interval $[0, c_1]$,there exists at least one $c \in (0, c_1)$ such that $f^{\prime \prime}(c)=0$. Since $(0, c_1) \subset (0, 1)$,there exists some $c \in (0, 1)$ such that $f^{\prime \prime}(c)=0$.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice continuously differentiable function such that $f(0)=f(1)=f^{\prime}(0)=0$. Then:

Similar Questions

If the function $f(x) = 2x^2 + 3x + 5$ satisfies the Lagrange's Mean Value Theorem $(LMVT)$ at $x = 3$ on the closed interval $[1, a]$,then the value of $a$ is equal to:

Consider the function $f(x) = |x - 2| + |x - 5|, x \in R$.
Statement-$1$: $f'(4) = 0$.
Statement-$2$: $f$ is continuous in $[2, 5]$,differentiable in $(2, 5)$ and $f(2) = f(5)$.

Consider $f(x) = |1 - x|$ for $1 \le x \le 2$ and $g(x) = f(x) + b \sin(\frac{\pi}{2}x)$ for $1 \le x \le 2$. Which of the following is correct?

If the function $f(x) = ax^3 + bx^2 + 11x - 6$ satisfies the conditions of Rolle's theorem for the interval $[1, 3]$ and $f'\left( 2 + \frac{1}{\sqrt{3}} \right) = 0$,then the values of $a$ and $b$ are respectively

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

Vedclass Test Series

Exam Paper Generator

Online Exam Module