(B) Given $f^{\prime}(x) = [f(x)]^2 + 4$. Since $f(x)$ is continuous on $[1, 3]$ and differentiable on $(1, 3)$,by the Mean Value Theorem,there exists

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(B) Given $f^{\prime}(x) = [f(x)]^2 + 4$. Since $f(x)$ is continuous on $[1, 3]$ and differentiable on $(1, 3)$,by the Mean Value Theorem,there exists at least one $c \in (1, 3)$ such that $f^{\prime}(c) = \frac{f(3)-f(1)}{3-1} = \frac{f(3)-f(1)}{2}$. Substituting this into the given differential equation: $\frac{f(3)-f(1)}{2} = [f(c)]^2 + 4$. This implies $f(3)-f(1) = 2[f(c)]^2 + 8$. Since $[f(c)]^2 \ge 0$,it follows that $f(3)-f(1) \ge 8$. Therefore,the statement $f(3)-f(1)=5$ cannot hold.

Class 12 Mathematics Continuity and Differentiation Rolle’s theorem, Lagrange's mean value theorem

Medium

Let $f:[1,3] \rightarrow R$ be continuous and differentiable in $(1,3)$ such that $f^{\prime}(x)=[f(x)]^2+4$ for all $x \in (1,3)$. Then:

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A
$f(3)-f(1)=5$ holds
B
$f(3)-f(1)=5$ does not hold
C
$f(3)-f(1)=3$ holds
D
$f(3)-f(1)=4$ holds

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Continuity and Differentiation

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Rolle’s theorem, Lagrange's mean value theorem

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Consider the function $f(x) = |x - 2| + |x - 5|$,$x \in R$.
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MediumAIEEE 2012

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Let $f$ be a function which is differentiable for all real $x$. If $f(2) = -4$ and $f^{\prime}(x) \geq 6$ for all $x \in [2, 4]$,then which of the following is true?

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Let $f:[1,3] \rightarrow R$ be continuous and differentiable in $(1,3)$ such that $f^{\prime}(x)=[f(x)]^2+4$ for all $x \in (1,3)$. Then:

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Verify the Mean Value Theorem for the function $f(x) = x^{3} - 5x^{2} - 3x$ in the interval $[1, 3]$. Find all $c \in (1, 3)$ such that $f^{\prime}(c) = \frac{f(3) - f(1)}{3 - 1}$.

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Let $a > 0$ and $f$ be continuous in $[-a, a]$. Suppose that $f'(x)$ exists and $f'(x) \le 1$ for all $x \in (-a, a)$. If $f(a) = a$ and $f(-a) = -a$,then $f(0)$ is:

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 $f(2) = f(5)$.

Let $f$ be a function which is differentiable for all real $x$. If $f(2) = -4$ and $f^{\prime}(x) \geq 6$ for all $x \in [2, 4]$,then which of the following is true?

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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 $f(2) = f(5)$.