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If f(x) = b\[{{\text{e}}^{{\text{ax}}}}\]+ a\[{{\text{e}}^{{\text{bx}}}}\], then ${{\text{f}}^{''}}$(0) =
  {\text{A}}{\text{. 0}} \\
  {\text{B}}{\text{. 2ab}} \\
  {\text{C}}{\text{. ab(a + b)}} \\
  {\text{D}}{\text{. ab}} \\

Last updated date: 24th Jul 2024
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Hint - To solve this question we differentiate the equation f(x) twice. Then we substitute 0 in place of x to determine the answer.

Complete step-by-step answer:
Given f(x) = b\[{{\text{e}}^{{\text{ax}}}}\]+ a\[{{\text{e}}^{{\text{bx}}}}\]
⟹f’(x) = $\dfrac{{{\text{df}}}}{{{\text{dx}}}}$ = b\[{{\text{e}}^{{\text{ax}}}}\]$\dfrac{{\text{d}}}{{{\text{dx}}}}$(ax) +a\[{{\text{e}}^{{\text{bx}}}}\]$\dfrac{{\text{d}}}{{{\text{dx}}}}$(bx) = b\[{{\text{e}}^{{\text{ax}}}}\]a +a\[{{\text{e}}^{{\text{bx}}}}\]b = ab (\[{{\text{e}}^{{\text{ax}}}}\]+ \[{{\text{e}}^{{\text{bx}}}}\])
($\dfrac{{\text{d}}}{{{\text{dx}}}}\left( {{{\text{e}}^{\text{x}}}} \right) = {{\text{e}}^{\text{x}}}$)
⟹f “(x) = $\dfrac{{{{\text{d}}^2}{\text{f}}}}{{{\text{d}}{{\text{x}}^2}}}$ = ab ( a\[{{\text{e}}^{{\text{ax}}}}\] + \[{{\text{e}}^{{\text{bx}}}}\]b)

⟹f “(0) = ab(${{\text{e}}^0}$a+${{\text{e}}^0}$b) (${{\text{e}}^0}$=1)

⟹f”(0) = ab (a+b)

Note: In order to solve this type of question the key is to carefully differentiate the equation.
Differentiation is a process of finding a function that outputs the rate of change of one variable with respect to another variable.
The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus.