JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let the function, \(f:[-7,0] \rightarrow R\) be continuous on \([-7,0]\) and differentiable on \((-7,0)\) If \(f(-7)=-3\) and \(f(x) \leq 2,\) for all \(x \in(-7,0),\) then for all such functions \(f, f(-1)+f(0)\) lies in the interval
- A \([-6,20]\)
- B \((-\infty, 20]\)
- C \((-\infty, 11]\)
- D \([-3,11]\)
Answer & Solution
Correct Answer
(B) \((-\infty, 20]\)
Step-by-step Solution
Detailed explanation
Using LMVT in \([-7,-1]\) \(\frac{f(-1)-f(-7)}{-1-(-7)} \leq 2\) \(f(-1)-f(-7) \leq 12\) \(\Rightarrow f(-1) \leq 9\) Using LMVT in \([-7,0]\) \(\frac{f(0)-f(-7)}{0-(-7)} \leq 2\) \(f(0)-f(-7) \leq 14\) \(f(0) \leq 11\) from ( 1) and ( 2) \(f(0)+f(-1) \leq 20\)
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