JAMIA 2025 — Mathematics PYQ
JAMIA | Mathematics | 2025The function f(x)=x3−3x is:
Choose the correct answer:
- A.
Increasing anywhere
- B.
Decreasing everywhere
- C.
Increasing in some intervals, decreasing in others
(Correct Answer) - D.
None of these
Increasing in some intervals, decreasing in others
Explanation
1. Nature of Function (Odd or Even)
Pehle check karte hain ki function Odd hai ya Even:
Nishkarsh: Yeh ek Odd Function hai aur origin ke symmetric hai.
2. Critical Points (Increasing/Decreasing)
Iska derivative nikalne par:
Critical points ke liye f′(x)=0 rakhte hain:
-
Increasing: Jab x < -1 aur x > 1 ho (f'(x) > 0).
-
Decreasing: Jab -1 < x < 1 ho (f'(x) < 0).
3. Maxima and Minima
Double derivative test se:
-
x=−1 par: f''(-1) = -6 < 0 (Local Maxima)
-
x=1 par: f''(1) = 6 > 0 (Local Minima)
Explanation
1. Nature of Function (Odd or Even)
Pehle check karte hain ki function Odd hai ya Even:
Nishkarsh: Yeh ek Odd Function hai aur origin ke symmetric hai.
2. Critical Points (Increasing/Decreasing)
Iska derivative nikalne par:
Critical points ke liye f′(x)=0 rakhte hain:
-
Increasing: Jab x < -1 aur x > 1 ho (f'(x) > 0).
-
Decreasing: Jab -1 < x < 1 ho (f'(x) < 0).
3. Maxima and Minima
Double derivative test se:
-
x=−1 par: f''(-1) = -6 < 0 (Local Maxima)
-
x=1 par: f''(1) = 6 > 0 (Local Minima)