NCERT Class 12 · Ch 6 Miscellaneous Exercise · Q13
NCERT Class 12 Maths · Chapter 6 Applications Of Derivatives · Miscellaneous Exercise Question 13
Let f be defined on [a,b], with f′(x)>0 for every x∈(a,b). Prove f increases on (a,b).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: Strictly increasing on (a,b).
- Choose arbitrary a<u<v<b. Differentiability on (a,b) implies continuity on the compact subinterval [u,v].
- By the mean value theorem, f(v)−f(u)=f′(ξ)(v−u)>0 for some ξ∈(u,v).
- Hence u<v implies f(u)<f(v). No continuity at the outer endpoints a,b is needed for this open-interval conclusion.
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