Mathematics - Application of Derivative Question with Solution | TestHub

MathematicsApplication of DerivativeMonotonicity-Rolle's-theroemMedium2 minQB

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Passage / Comprehension

Let $f (x)$ be differentiable wherever it is continuous, with $f^{'} (c_{1}) = f^{'} (c_{2}) = 0, f^{''} (c_{1}) f^{''} (c_{2}) < 0$ , $f (c_{1}) = 5, f (c_{2}) = 0$ and $c_{1} < c_{2}$ .

If $f (x)$ is continuous in $[c_{1}, c_{2}]$ and $f^{''} (c_{1}) - f^{''} (c_{2}) > 0$ , then the minimum number of roots of $f (x) = 0$ in $[c_{1} - 1, c_{2} + 1]$ is

Options:

Answer:

Solution:

As in the first case, $c_{1}$ is a local minimum with $f (c_{1}) = 5 > 0$ and $c_{2}$ is a local maximum with $f (c_{2}) = 0$ . Just left of $c_{2}$ we have $f < 0$ , while $f (c_{1}) = 5 > 0$ , so $f$ has a root in $(c_{1}, c_{2})$ by continuity. Together with the root $x = c_{2}$ , there are at least 2 roots; the minimum is 2.

Stream:JEESubject:MathematicsTopic:Application of DerivativeSubtopic:Monotonicity-Rolle's-theroem

⏱ 2mℹ️ Source: QB

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