Determining where the function is concave up? (Given an integral)

[JessicaJ283782]

Homework Statement

x
Determine all intervals over which the function f(x)=∫√(1+t^2) dt is concave upward
1

Homework Equations

I know concave up means the f"(x)>0, so you have to get the second derivative

The Attempt at a Solution

f'(x)=√(1+x^) by susbituting in x

Then f"(x)=1/2*(1+x^2)^(-1/2)*2x

This is equal to:
2x/2√(1+x^2)

Then, I'm really confused on what to do next because the denominator is never zero, so would the only critical point be 0 and you check to the left and right of that?

 

[LCKurtz]

Homework Statement

x
Determine all intervals over which the function f(x)=∫√(1+t^2) dt is concave upward
1

Apparently that is ##\int_1^x \frac 1 {\sqrt{1+t^2}}~dt##

Homework Equations

I know concave up means the f"(x)>0, so you have to get the second derivative

The Attempt at a Solution

f'(x)=√(1+x^) by susbituting in x

Then f"(x)=1/2*(1+x^2)^(-1/2)*2x

This is equal to:
2x/2√(1+x^2)

Then, I'm really confused on what to do next because the denominator is never zero, so would the only critical point be 0 and you check to the left and right of that?

Yes. Find where it is positive. Don't be afraid to cancel the 2's.

 

[JessicaJ283782]

I apologize for the typo! So it would be:

x/√(1+x^2)

The only critical number would be 0 then, so wouldn't it be concave up on (-infinity, 0) and (0, infinity) since the denominator is always positive because of the square root?

 

[LCKurtz]

Is the numerator always positive?

 

[JessicaJ283782]

The numerator would give only a critical point of 0 since that is the only place it is not defined, right? So then it would be concave up on (0, infinity)?

 

[LCKurtz]

There is no ##x## where that fraction is not defined. Instead of asking me if ##(0,\infty)## is the answer, explain why you think it is.

 

[JessicaJ283782]

It would be (0, ∞) since the critical point is 0 because it makes the fraction zero. Concave up means f"(x)>0 and f"(x) is positive when the function is evaluated at any number greater than 0. If you evaluate the function at -1, for example, you would get a negative number, so it would be concave down less than 0. If that makes sense?

 

[LCKurtz]

It would be (0, ∞) since the critical point is 0 because it makes the fraction zero. Concave up means f"(x)>0 and f"(x) is positive when the function is evaluated at any number greater than 0. If you evaluate the function at -1, for example, you would get a negative number, so it would be concave down less than 0. If that makes sense?

Yes, that makes perfect sense. Since ##f''(x)=0## when ##x=0##, that is a point where the second derivative might change sign (or maybe not). That's why you check the signs on both sides. For ##x<0## you have ##\frac - +## and for ##x>0## you have ##\frac + +## for the signs. That's why ##f''(x)>0## for ##x>0##. That's how you want to analyze problems like this.

 

[JessicaJ283782]

Thank you so much! I really appreciate it!