Let the function, f: [-7,0] → R be continuous on [−7,0] and differentiable on (-7,0). If f(-7) = -3 and f'(x) ≤ 2, for all x ∈ (−7,0), then for all such functions f, f(−1) + f(0) lies in the interval:
Let the function, f: [-7,0] → R be continuous on [−7,0] and differentiable on (-7,0). If f(-7) = -3 and f'(x) ≤ 2, for all x ∈ (−7,0), then for all such functions f, f(−1) + f(0) lies in the interval:
Option 1 -
[-3,11]
Option 2 -
(-∞,20]
Option 3 -
[-6,20]
Option 4 -
(-∞, 11]
Asked by Shiksha User
-
1 Answer
-
Correct Option - 4
Detailed Solution:Using Lagrange's Mean Value Theorem (LMVT) for x ∈ [−7, -1].
[f (-1) - f (-7)] / [-1 - (-7)] ≤ 2
[f (-1) - (-3)] / 6 ≤ 2
f (-1) + 3 ≤ 12
f (-1) ≤ 9Using LMVT for x ∈ [−7, 0].
[f (0) - f (-7)] / [0 - (-7)] ≤ 2
[f (0) - (-3)] / 7 ≤ 2
f (0) + 3 ≤ 14
f (0) ≤ 11Therefore, f (0) + f (-1) ≤ 11 + 9 = 20.
Taking an Exam? Selecting a College?
Get authentic answers from experts, students and alumni that you won't find anywhere else
Sign Up on ShikshaOn Shiksha, get access to
- 65k Colleges
- 1.2k Exams
- 688k Reviews
- 1800k Answers