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Application of Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives1 Mark
MCQ
The number which exceeds its square by the greatest possible quantity is,
  • ✓
    $\frac{1}{2}$
  • B
    $\frac{1}{4}$
  • C
    $\frac{3}{4}$
  • D
    None of these.
✓

Answer

Correct option: A.
$\frac{1}{2}$
Let the required number be $x.$Then, $f^{\prime}(x)=x-x^2$
lmplies that $f(x) = 1 - 2x = 0$
For a local maxima or a local minima, we must have $f\ '(x) = 0$
lmplies that $2x = 1$
lmplies that $\text{x}=\frac{1}{2}$
Now, $f\ ''(x) = -2 < 0$
Therefore, $\text{x}=\frac{1}{2}$ is a local maxima.
Hence, the required number is $\frac{1}{2}$.

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