How do you find the critical numbers of #f(x)=(x^2)(e^(11x))#?

Question

Christopher Allers · Accepted Answer

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# x \ = \ 0, - 2/11. # # \ # # \mbox{The critical numbers} \ \ f(x) \ \ \mbox{are the solutions of:} # # \qquad \qquad \qquad \qquad \ \ f'(x) \ = \ 0 \ \ \mbox{and} \ \ f'(x) \ = \ \mbox{undefined} . # # \mbox{So the first step will be to calculate} \ f'(x), \mbox{and to do this, we} \ \mbox{will use the product rule as the main part of the computation. # # \mbox{1) Given:} \qquad \qquad \qquad \qquad \qquad f(x) \ = \ ( x^2 ) ( e^{ 11 x } ). # # \mbox{2) Product Rule:} \qquad \ \ f'(x) \ = \ ( x^2 ) [ ( e^{ 11 x } ) ]' \ + \ ( x^2 )' [ ( e^{ 11 x } ) ]. # # \mbox{3) Special Function Rules -- Power, Exponential Functions:} # # \qquad \qquad f'(x) \ = \ ( x^2 ) [( 11 x )'( e^{ 11 x } ) ] \ + \ ( 2 x ) [( e^{ 11 x } ) ]; # # \qquad \qquad f'(x) \ = \ 11 ( x^2 ) ( e^{ 11 x } ) \ + \ ( 2 x ) ( e^{ 11 x } ). # # \mbox{5) Simplification -- Factor Out Common Factor} \ \ e^{ 11 x } \mbox{:} # # \qquad \qquad f'(x) \ = \ [ 11 ( x^2 ) \ + \ ( 2 x ) ] ( e^{ 11 x } ); # # \qquad \qquad f'(x) \ = \ (11 x^2 \ + \ 2 x ) e^{ 11 x }. # # \mbox{4) Critical Numbers:} # # \qquad \qquad \mbox{a) Solve:} \ \ f'(x) \ = \ 0. # # \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (11 x^2 \ + \ 2 x ) e^{ 11 x } \ = \ 0 # # \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (11 x^2 \ + \ 2 x ) e^{ 11 x } \cdot e^{ -11 x } \ = \ 0 \cdot e^{ -11 x } # # \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (11 x^2 \ + \ 2 x ) e^{0 } \ = \ 0 # # \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (11 x^2 \ + \ 2 x ) cdot 1 \ = \ 0 # # \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 11 x^2 \ + \ 2 x \ = \ 0 # # \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x (11 x \ + \ 2 ) \ = \ 0 # # \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x \ = \ 0 \qquad \mbox{or} \qquad x \ = \ - 2/11. # # \qquad \qquad \mbox{b) Solve:} \ f'(x) = \mbox{undefined. (Where is} \ f'(x) \ \mbox{undefined ?)} # # \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (11 x^2 \ + \ 2 x ) e^{ 11 x } = \mbox{undefined} # # \qquad \qquad \mbox{Clearly,} \ (11 x^2 \ + \ 2 x ) e^{ 11 x } \ \mbox{is defined everywhere.} # # \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mbox{So no solutions here here.} # # \qquad \qquad \mbox{c) Combine solutions from parts (a) and (b):} # # \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x \ = \ 0, - 2/11. # # \qquad \qquad \qquad \qquad \qquad \qquad \mbox{These are our critical points.} # # \qquad \qquad \mbox{d) State Solution:} # # \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad f(x) \ = \ ( x^2 ) ( e^{ 11 x } ). # # \mbox{Critical points:} \qquad \qquad \qquad x \ = \ 0, - 2/11. #

Christopher Allers · Answer

undefined To find the critical numbers of $ f(x) = x^2 \cdot e^{11x} $, you need to find the values of $ x $ where the derivative of $ f(x) $ is equal to zero or undefined. Here's the process:

1. Find the first derivative of $ f(x) $ using the product rule.
2. Set the first derivative equal to zero and solve for $ x $.
3. Check for any values of $ x $ where the first derivative is undefined (usually where the function is not differentiable).

Once you've found the critical numbers, you can analyze them further for maxima, minima, or points of inflection if needed.