UPSC NDA I 2025 Mathematics Paper

Given,
The function is f(x)=x2+9.
Statement I: f(x) is an increasing function.
The derivative of f(x) is:
f′(x)=‌

d

dx

(x2+9)=2x
When (x>0),(f′(x)>0), so (f(x) is increasing.
When (x<0),f′(x)<0 ), so f(x) is decreasing.
At (x=0),(f′(x)=0), meaning the function is neither increasing nor decreasing at this point.
Hence, f(x) is not entirely increasing. It is increasing for ( x>0 ) and decreasing for ( x<0 ).
Statement II: f(x) has local maximum at x=0
Since the function f(x)=x2+9 is a parabola opening upwards (because the coefficient of x2 is positive), it has a global minimum at x=0, not a local maximum.
Conclusion:
- Statement I is incorrect because the function is not entirely increasing. It is increasing for x>0 and decreasing for x<0.
- Statement II is incorrect because the function has a global minimum at x=0, not a local maximum.