If 𝑓(𝑥+𝑦)=𝑓(𝑥)𝑓(𝑦) for all 𝑥,𝑦∈R and 𝑓(5)=2,𝑓^′ (0)=3, then

Question 24 If 𝑓(𝑥+𝑦)=𝑓(𝑥)𝑓(𝑦) for all 𝑥,𝑦∈R and 𝑓(5)=2,𝑓^′ (0)=3, then using the definition of derivatives, find 𝑓^′ (5).When it says using the definition of derivatives, it means using the formula f’(x) = (𝒍𝒊𝒎)┬(𝒉→𝟎)⁡〖(𝒇(𝒙 + 𝒉) − 𝒇 (𝒙))/𝒉〗 Putting x = 5 f’(5) = (𝑙𝑖𝑚)┬(ℎ→0)⁡〖(𝑓(5 + ℎ) − 𝑓(5))/ℎ〗 Using 𝑓(𝑥+𝑦)=𝑓(𝑥)𝑓(𝑦) f’(5) = (𝑙𝑖𝑚)┬(ℎ→0)⁡〖(𝑓(5) × 𝑓(ℎ) − 𝑓(5))/ℎ〗 f’(5) = (𝒍𝒊𝒎)┬(𝒉→𝟎)⁡〖(𝒇(𝟓) [𝒇(𝒉) − 𝟏])/𝒉〗 Putting f(5) = 2 f’(5) = (𝑙𝑖𝑚)┬(ℎ→0)⁡〖(2 × [𝑓(ℎ) − 1])/ℎ〗 f’(5) = 𝟐 (𝒍𝒊𝒎)┬(𝒉→𝟎)⁡〖[𝒇(𝒉) − 𝟏]/𝒉〗 Now, since f’(x) = (𝑙𝑖𝑚)┬(ℎ→0)⁡〖(𝑓(𝑥 + ℎ) − 𝑓 (𝑥))/ℎ〗 Putting x = 0 f’(0) = (𝑙𝑖𝑚)┬(ℎ→0)⁡〖(𝑓(0 + ℎ) − 𝑓(0))/ℎ〗 f’(0) = (𝑙𝑖𝑚)┬(ℎ→0)⁡〖(𝑓(ℎ) − 𝑓(0))/ℎ〗 Putting f’(0) = 3 3 = (𝑙𝑖𝑚)┬(ℎ→0)⁡〖(𝑓(ℎ) − 𝑓(0))/ℎ〗 (𝒍𝒊𝒎)┬(𝒉→𝟎)⁡〖(𝒇(𝒉) − 𝒇(𝟎))/𝒉〗 = 3 Putting (2) in (1) f’(5) = 2 (𝑙𝑖𝑚)┬(ℎ→0)⁡〖[𝑓(ℎ) − 1]/ℎ〗 f’(5) = 2 × 3 f’(5) = 𝟔

Question 24 - CBSE Class 12 Sample Paper for 2026 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

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