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Calculus Level pending

Suppose the f u n c t i o n function f ( x ) − f ( 2 x ) f(x) - f(2x) has the d e r i v a t i v e derivative 5 5 at x = 1 x = 1 and d e r i v a t i v e derivative 7 7 at x = 2 x = 2 . The d e r i v a t i v e derivative 0f the f u n c t i o n function f ( x ) − f ( 4 x ) f(x) - f(4x) at x = 1 x = 1 ; has value equals to ? ?


The answer is 19.

Harsh Soni by harsh soni , 23, India

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1 solution

g ( x ) = [ f ( x ) − f ( 2 x ) ] ′ = f ′ ( x ) − 2 f ′ ( 2 x ) ⇒ { g ( 1 ) = f ′ ( 1 ) − 2 f ′ ( 2 ) = 5 g ( 2 ) = f ′ ( 2 ) − 2 f ′ ( 4 ) = 7 h ( x ) = [ f ( x ) − f ( 2 x ) ] ′ = f ′ ( x ) − 4 f ′ ( 4 x ) ⇒ h ( 1 ) = f ′ ( 1 ) − 4 f ′ ( 4 ) = g ( 1 ) + 2 g ( 2 ) = 5 + 2 × 7 = 19 \begin{array}{l} g(x) = [f(x) - f(2x)]' = f'(x) - 2f'(2x) \\ \Rightarrow \left\{ \begin{array}{l} g(1) = f'(1) - 2f'(2) = 5 \\ g(2) = f'(2) - 2f'(4) = 7 \\ \end{array} \right. \\ h(x) = [f(x) - f(2x)]' = f'(x) - 4f'(4x) \\ \Rightarrow h(1) = f'(1) - 4f'(4) = g(1) + 2g(2) = 5 + 2 \times 7 = \boxed{19} \\ \end{array}

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