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

Answer 1

#xe^{15x} ( 2+15x)#

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:

#(d/dx x^2) * e^{15x} + x^2 * (d/dx e^{15x})#
#2x * e^{15x} + x^2 * (d/dx e^{15x})#
#d/dx e^{15x} = e^{15x} * (d/dx 15x) = e^{15x} * 15#

So, the answer is

#2x * e^{15x} + 15x^2 e^{15x}#
We can factor an exponential and a #x#, obtaining
#xe^{15x} ( 2+15x)#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

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} ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7