Suppose f''(x) exists for all real x. If f(2) | vedclass

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

Question

(B) Let $f(x)$ be a function such that $f(2)=2, f(3)=5, f(4)=10$. By Lagrange's Mean Value Theorem on $[2, 3]$,there exists $c_1 \in (2, 3)$ such that

Vedclass · Accepted Answer

$f''(x) > 1$ for some $x \in (2, 4)$

Vedclass · Answer

(B) Let $f(x)$ be a function such that $f(2)=2, f(3)=5, f(4)=10$. By Lagrange's Mean Value Theorem on $[2, 3]$,there exists $c_1 \in (2, 3)$ such that $f'(c_1) = \frac{f(3)-f(2)}{3-2} = \frac{5-2}{1} = 3$. By Lagrange's Mean Value Theorem on $[3, 4]$,there exists $c_2 \in (3, 4)$ such that $f'(c_2) = \frac{f(4)-f(3)}{4-3} = \frac{10-5}{1} = 5$. Now,applying the Mean Value Theorem to $f'(x)$ on the interval $[c_1, c_2]$,there exists $c \in (c_1, c_2) \subset (2, 4)$ such that $f''(c) = \frac{f'(c_2)-f'(c_1)}{c_2-c_1} = \frac{5-3}{c_2-c_1} = \frac{2}{c_2-c_1}$. Since $c_1 \in (2, 3)$ and $c_2 \in (3, 4)$,the length of the interval $c_2-c_1$ is less than $2$. Specifically,$0 Therefore,$f''(c) = \frac{2}{c_2-c_1} > \frac{2}{2} = 1$. Thus,$f''(x) > 1$ for some $x \in (2, 4)$.

Suppose $f''(x)$ exists for all real $x$. If $f(2) = 2$,$f(3) = 5$ and $f(4) = 10$,then which one among the following statements is definitely true?

Similar Questions

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:

Consider the function $f(x) = 8x^2 - 7x + 5$ on the interval $[-6, 6]$. The value of $c$ that satisfies the conclusion of the Mean Value Theorem is:

If Rolle's theorem holds for the function $f(x)=x^{3}-ax^{2}+bx-4$ on the interval $x \in [1, 2]$ with $f^{\prime}\left(\frac{4}{3}\right)=0$,then the ordered pair $(a, b)$ is equal to

Let $f$ be continuous on $[1, 5]$ and differentiable in $(1, 5).$ If $f(1)=-3$ and $f'(x) \ge 9$ for all $x \in (1, 5)$,then which of the following is true?

Let $f(x)=2+\cos x$ for all real $x$.
$STATEMENT-1$: For each real $t$,there exists a point $c$ in $[t, t+\pi]$ such that $f^{\prime}(c)=0$. because
$STATEMENT-2$: $f(t)=f(t+2\pi)$ for each real $t$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module