Integral Calculus
Integral Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
If $g(x) = \int_0^x 2|t| dt$, then:
$g(x) = x|x|$
$g(x)$ is monotonic
$g(x)$ is differentiable at $x = 0$
$g'(x)$ is differentiable at $x = 0$
Step-by-Step Solution
Key Concept: Piecewise absolute value functions become non-differentiable at transition points where the derivative has a jump discontinuity.
The function $g(x) = 2|\int_0^x t\,dt|$ is evaluated by splitting into cases: for $x 0$, making it non-differentiable at the origin.
Correct Answer: 2,3