# What are the critical values, if any, of #f(x) = f(x) = x^{2}e^{15 x}#?

To find the critical points, we need the first derivative. This function is a multiplication of a power and a composite exponential. Let's see how to deal with these three things:

Let's put all these things together:

So, the answer is

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To find the critical values of ( f(x) = x^2e^{15x} ), we need to find where the derivative of ( f(x) ) equals zero or is undefined.

The derivative of ( f(x) ) can be found using the product rule:

( f'(x) = (2x)(e^{15x}) + (x^2)(15e^{15x}) )

Setting ( f'(x) ) equal to zero and solving for ( x ), we have:

( (2x)(e^{15x}) + (x^2)(15e^{15x}) = 0 )

( e^{15x}(2x + 15x^2) = 0 )

This equation will be zero if either ( e^{15x} = 0 ) or ( 2x + 15x^2 = 0 ).

However, ( e^{15x} ) is never zero for any real number ( x ), so we only need to solve for ( 2x + 15x^2 = 0 ).

Factoring out ( x ), we get:

( x(2 + 15x) = 0 )

This equation is satisfied if ( x = 0 ) or ( 2 + 15x = 0 ).

Solving ( 2 + 15x = 0 ) for ( x ), we find:

( 15x = -2 )

( x = -\frac{2}{15} )

So, the critical values of ( f(x) = x^2e^{15x} ) are ( x = 0 ) and ( x = -\frac{2}{15} ).

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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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