For what values of x is # f(x) = e^(4x)-4e^(2x)+e^x # concave or convex?

Answer 1

Concave: #(-∞,-2.769)∪(-0.03,+∞)#
Convex: #(-2.769,-0.03)#

First we need to calculate the second derivate of the function, #f''(x)=16e^(4x)-16e^(2x)+e^x#
then we equal it to #0#, #16e^(4x)-16e^(2x)+e^x=0#
and we solve for #x#, #x_1~=-2.769# #x_2~=-0.03#
and we substitute the second derivate by a number between these intervals: #(-∞,-2.769)# and #(-2.769,-0.03)# and #(-0.03,+∞)#. If the number we get is negative it means that the function is convex in that interval, if it's possitive means that's concave, #f''(-10)=4.54·10^-5# #f''(-1)=-1.5# #f''(5)=7762290852#
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Answer 2

To determine the concavity of the function ( f(x) = e^{4x} - 4e^{2x} + e^x ), we need to find the second derivative of the function and then analyze its sign.

First, find the first derivative of ( f(x) ):

[ f'(x) = 4e^{4x} - 8e^{2x} + e^x ]

Now, find the second derivative of ( f(x) ) by differentiating ( f'(x) ):

[ f''(x) = 16e^{4x} - 16e^{2x} + e^x ]

Now, to analyze the concavity of ( f(x) ), we need to examine the sign of ( f''(x) ) for different values of ( x ).

  • If ( f''(x) > 0 ) for a certain interval, then ( f(x) ) is concave up on that interval.
  • If ( f''(x) < 0 ) for a certain interval, then ( f(x) ) is concave down on that interval.

To find the intervals where ( f(x) ) is concave up or concave down, you'll need to solve the inequality ( f''(x) > 0 ) or ( f''(x) < 0 ), respectively.

[ 16e^{4x} - 16e^{2x} + e^x > 0 ]

This inequality may not have a straightforward solution without using numerical methods or further simplification. However, if you're looking for specific points of concavity change (inflection points), you would set ( f''(x) = 0 ) and solve for ( x ).

[ 16e^{4x} - 16e^{2x} + e^x = 0 ]

Solving this equation will give you the x-values where the concavity of the function may change.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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