For what values of x is \(\displaystyle{f{{\left({x}\right)}}}={\frac{{{4}}}{{{x}^{{{2}}}}}}+{1}\)

Polchinio3es

Polchinio3es

Answered question

2022-03-27

For what values of x is f(x)=4x2+1 concave or convex?

Answer & Explanation

Kathleen Sanchez

Kathleen Sanchez

Beginner2022-03-28Added 7 answers

Step 1
The determine when a function is concave or convex, analyze the sign, positive or negative, of the function's second derivative:
When f>0, then f is convex.
When f<0, then f is concave.
So, we first must find f
Note that we can write f as
f(x)=4x2+1
Now, through the power rule, we see that
f(x)=8x3
f(x)=24x4=24x4
We must now determine when 24x4 is positive or negative.
It's necessary to note that x4 will always be positive, so 24x4 will also always be positive. Recall that the domain of f excludes 0, so we know that
f is convex on the interval (, 0)(0, +)

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