Limits, Continuity & Differentiability
Continuity And Differentiability
nta_abhyas_2025
Grade 12
Question:
If $f(x) = |x|^{\{x\}} + (\{x\})^2 + \sin(\pi x)$, where $[\cdot]$ and $\{\cdot\}$ represent the greatest integer function and the fractional part function respectively, then $f'\left(\frac{3}{2}\right)$ is
$\sqrt{3} \ln 3 + \frac{\pi}{2}$
$\sqrt[3]{3} + \frac{3\pi}{2}$
$\sqrt[3]{3} + \pi + \frac{1}{4}$
$\sqrt[3]{3} + \frac{\pi}{2}$
Step-by-Step Solution
Key Concept: Differentiate piecewise and composite functions carefully, ensuring proper handling of absolute values and fractional exponents.
$f(x) = |x|^3 + |x|^{2/3} + \sin(\pi x)$ with domain $(3, 4)$. Computing the derivative: $f'(x) = 3x^2 + 3(x-3)^{-2/3} + \pi\cos(\pi x)$. At $x = \frac{7}{3}$: $f'(\frac{7}{3}) = \sqrt{3}\ln 3 + \frac{\pi}{4}$.
Correct Answer: $\sqrt{3}ln 3 + \frac{\pi}{4}$