Let f: R rightarrow R be a twice continuously | vedclass

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

Question

(A) Given that $f(0) = f(1) = 0$. By Rolle's theorem,there exists at least one point $c \in (0, 1)$ such that $f^{\prime}(c) = 0$. We are also given t

Vedclass · Accepted Answer

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

Vedclass · Answer

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

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

Similar Questions

If a polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \dots + a_2x^2 + a_1x + a_0 = 0$,where $n$ is a positive integer,has two distinct roots $\alpha$ and $\beta$,then how many roots does the equation $na_nx^{n-1} + (n - 1)a_{n-1}x^{n-2} + \dots + a_1 = 0$ have in the interval $(\alpha, \beta)$?

If Rolle's theorem holds for the function $f(x) = 2x^3 + bx^2 + cx$ on the interval $x \in [-1, 1]$ at the point $x = \frac{1}{2}$,then the value of $2b + c$ is:

For the Mean Value Theorem $f(b) - f(a) = (b - a) f'(x_1)$ where $a < x_1 < b$,if $f(x) = 1/x$,then $x_1 = ?$

For the function $f(x) = e^{\cos x}$,Rolle's theorem is

Let $y = f(x)$ and $y = g(x)$ be two differentiable functions in $[0, 2]$ such that $f(0) = 3$,$f(2) = 5$,$g(0) = 1$,and $g(2) = 2$. If there exists at least one $c \in (0, 2)$ such that $f'(c) = k g'(c)$,then $k$ must be:

Vedclass Test Series

Exam Paper Generator

Online Exam Module