# What are the critical points of #f(x) = x/(e^(sqrtx)#?

Use the quotient rule and the definition of the derivative of the

To find the critical points, we set this equal to 0...

graph{x/(e^sqrtx) [-10, 10, -5, 5]}

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

To find the critical points of ( f(x) = \frac{x}{e^{\sqrt{x}}} ), we first find the derivative of ( f(x) ) and then set it equal to zero to solve for ( x ). The derivative is ( f'(x) = \frac{e^{\sqrt{x}} - \frac{x}{2\sqrt{x}}e^{\sqrt{x}}}{e^{2\sqrt{x}}} ). Setting this equal to zero and simplifying gives us ( e^{\sqrt{x}}(2 - \frac{x}{2\sqrt{x}}) = 0 ). This simplifies to ( 2e^{\sqrt{x}} - \frac{x}{\sqrt{x}}e^{\sqrt{x}} = 0 ), which further simplifies to ( 2e^{\sqrt{x}} - xe^{\sqrt{x}} = 0 ). Factoring out an ( e^{\sqrt{x}} ) gives ( e^{\sqrt{x}}(2 - x) = 0 ). Setting each factor to zero gives us ( e^{\sqrt{x}} = 0 ) (which has no real solutions) and ( 2 - x = 0 ), which gives ( x = 2 ). Therefore, the critical point of ( f(x) ) is ( x = 2 ).

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

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.

- What is the minimum value of #f(x)=3x^2-6x+12#?
- Is #f(x)= cos(x+(pi)/4) # increasing or decreasing at #x=pi/3 #?
- How do you use the Intermediate Value Theorem to show that the polynomial function #f(x) = x^4 -10x^2 + 3# has one zero?
- Is #f(x)=cotx-e^xtanx# increasing or decreasing at #x=pi/6#?
- What are the values and types of the critical points, if any, of #f(x)=x^3-6x^2+12x-6#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7