Question

Help Please! (:
f(x)=x(sqrt(x^2+25)) is defined in an interval [-7,5]

1.) Find the concave up and down regions.
2.) What is the inflection point of the function.
3.) Where does the maximum and minimum occur.

Answer 1

So for minimum and maximum points, we need to set the first derivative equal to 0 and for inflection points we need to set the 2nd derivative = to 0. SO beginning with the first derivative, we have a product rule. So the product rule means we need
f'(x)g(x) + f(x)g'(x) where f(x) will equal x and g(x) will equal √(x^2+25). So doing this, we will get (1)(√(x^2+25)) + (x)(1/2)(x^2+25)^(-1/2)(2x). I can simplify this some to get √(x^2 + 25) + [x^2/(√(x^2+25))] To further make the second derivative simpler, I will multiply up the √(x^2+25) in the denominator of the fraction to make a common denominator as well as one fraction. Once I multiply it up, I have
([√(x^2+25)][√(x^2+25)] + x^2)/√(x^2+25). SImplifying this all the way I will have
(2x^2 + 25)/√(x^2+25). From this I can see if I have any critical points. In this case I do not because 2x^2 + 25 set to 0 would only give imaginary answers. Therefore the only points we can test for a minimum and a maximum are our interval points of -7 and 5. So plugging in -7 and 5 into the original equation, we have the coordinates -7√(74) and 5√(50), which you can simplify further if needed. But clearly your minimum point would occur at x = -7 and your maximum would be at x = 5.
So now we finally need our 2nd derivative. Using (2x^2 + 25)/(√(x^2+25)), we now want the quotient rule, which is (f'(x)g(x) - f(x)g'(x))/(g(x))^2. So I'll now do that. I will have [4x(√(x^2+25)) - [(2x+25)(1/2)(x^2+25)^(-1/2)(2x)]]/(x^2+25). Simplifying further I will get a pretty complicated stacking fraction. I can have [4x(√(x^2+25)) - {2x^3 + 25x)/(√(x^2+25))]/(x^2+25) Bit hard to have that all written out, but Ill further simplify now by making everything into a single fraction. After all the simplification, I have (2x^3+75x)/(x2+25)^(3/2).The numerator I can factor out an x to leave me with x(2x^2 + 75) in the numerator. In this case, the only real number inflection point is x = 0. So now in terms of concavity, I will pick 2 points, one on each side of the inflection point, to see how the function is behaving. If I plug in x = -1 into the 2nd deriviative, I'll get a negative answer (since I only care about the sign anyway), meaning on your interval from -7 to 0, the function is concave down. If I test x = 1, I end up with a positive answer meaning the function is concave up from 0 to 5. So a recap of everything:
There were no critical points, so we tested the interval end points for our minimum and maximum. This gave us our minimum at x = -7 and our max at x = 5. Taking the 2nd derivaitve we found our one inflection point at x = 0 and that from -7 to 0 we were concave down and from 0 to 5 we were concave up. I know this is a big wall of text, so see if any of it makes sense.