1. Function Definition
The function p(x) is defined as the product of f(x) and g(x):
p(x)=tan(x2)⋅x∣x∣
2. Checking Continuity at x=0
A function is continuous at x=a if limx→ap(x)=p(a).
First, evaluate p(0):
p(0)=tan(02)⋅0⋅∣0∣=0⋅0=0
Now, find the limit as x→0:
x→0lim(tan(x2)⋅x∣x∣)=x→0limtan(x2)⋅x→0lim(x∣x∣)=0⋅0=0
Since limx→0p(x)=p(0)=0, the function p(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 p′(a) exists, defined by:
p′(0)=h→0limhp(0+h)−p(0)
Substitute p(h)=tan(h2)⋅h∣h∣ and p(0)=0:
p′(0)=h→0limhtan(h2)⋅h∣h∣−0
p′(0)=h→0lim(tan(h2)⋅∣h∣)
As h→0, tan(h2)→0 and ∣h∣→0. Therefore:
p′(0)=0⋅0=0
Since the limit exists and equals 0, the function p(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
