1. Define the Composite Function q(x)
Given f(x)=tan(x2) and g(x)=x∣x∣, the composite function q(x)=f(g(x)) is:
q(x)=f(x∣x∣)=tan((x∣x∣)2)=tan(x2∣x∣2)
Since ∣x∣2=x2, we can simplify this to:
q(x)=tan(x4)
2. Checking Continuity at x=0
A function is continuous at x=a if limx→aq(x)=q(a).
First, evaluate q(0):
q(0)=tan(04)=tan(0)=0
Now, find the limit as x→0:
x→0limq(x)=x→0limtan(x4)=tan(0)=0
Since limx→0q(x)=q(0)=0, the function q(x) is continuous at x=0. Thus, Statement I is correct.
3. Checking Differentiability at x=0
A function is differentiable at x=a if the derivative q′(a) exists, defined by:
q′(0)=h→0limhq(0+h)−q(0)
Substitute q(h)=tan(h4) and q(0)=0:
q′(0)=h→0limhtan(h4)−0=h→0limhtan(h4)
Using the standard limit limθ→0θtanθ=1, we can rewrite the limit by multiplying and dividing by h4:
q′(0)=h→0lim(h4tan(h4)⋅h3)=1⋅0=0
Since the limit exists and equals 0, the function q(x) is differentiable at x=0. Thus, Statement II is correct.
Conclusion: Both Statement I and Statement II are correct.
Correct Option: (c) Both I and II
