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

##### 1 Answer

#### Explanation:

#f(x)=(e^(x^(1/2))-x^(1/2))^(1/2)#

Through the chain rule:

#f'(x)=1/2(e^(x^(1/2))-x^(1/2))^(-1/2)d/dx(e^(x^(1/2))-x^(1/2))#

Then:

#f'(x)=1/2(e^(x^(1/2))-x^(1/2))^(-1/2)(e^(x^(1/2))(1/2x^(-1/2))-1/2x^(-1/2))#

Factoring from the final parentheses:

#f'(x)=1/2(e^(x^(1/2))-x^(1/2))^(-1/2)(1/2x^(-1/2))(e^(x^(1/2))-1)#

Rewriting:

#f'(x)=1/(2(e^(x^(1/2))-x^(1/2))^(1/2)(2x^(1/2)))(e^(x^(1/2))-1)#

#f'(x)=(e^sqrtx-1)/(4sqrtxsqrt(e^sqrtx-sqrtx))#

If we want to find critical point, we need to find when

Setting

This is also when

Thus the only critical point is