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Mathematics - Application of Derivative Question with Solution | TestHub

MathematicsApplication of DerivativeMonotonicity-Rolle's-theroemMedium2 minQB

MathematicsMediumsingle choice

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

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Answer:

Solution:

Now $f^{''} (c_{1}) < 0 < f^{''} (c_{2})$ : local maximum at $c_{1}$ (value 5) and local minimum at $c_{2}$ (value 0). $f$ can decrease steadily from 5 to 0 between them, so no further root is forced. The minimum number of roots is 2 ( $c_{1}$ and $c_{2}$ ).

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

⏱ 2mℹ️ Source: QB

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