Math CentralQuandaries & Queries


Question from natalie, a student:

I want to graph the curve of y=(4-x^2)^5 without using a graphing calculator. To do this, I'm suppose to find: domain, y and x intercepts, asymptotes, intervals of increase/decrease, local max/min, concavity and points of inflection. I got all the way to the step where I'm solving the concavity and I'm stuck. I found the f"(x) and it came out to be really large polynomial. I want to know how I can solve for the x of f"(x) without the use of a graphing calculator, when the polynomial has x^6 and x^8.
Thank you so much,

Hi Natalie,

If you are ever called on to solve this type of problem in a practical situation the complex algebra would probably force you to use a graphing calculator or some technology that approximates the roots of a polynomial. This problem however is a "text-book problem" and set up so that you can find the roots of the second derivative.

f(x) = (4 - x2)5


f'(x) = 5(4 - x2)4(-2x) = -10x(4 - x2)4.

Differentiating a second time gives

f"(x) = -10(4 - x2)4 - 10x[4(4 - x2)3(-2x)]
= -10(4 - x2)4 + 80x2(4 - x2)3
= (4 - x2)3[-10(4 - x2) + 80x2]
= (4 - x2)3[-40 + 70x2]

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