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

Expert answers to "For what values of x is f(x)=4x2+1 concave..."

Polchinio3es

2022-03-27

For what values of x is

f (x) = \frac{4}{x^{2}} + 1

concave or convex?

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) = 4 x^{- 2} + 1

Now, through the power rule, we see that

f^{'} (x) = - 8 x^{- 3}

f^{″} (x) = 24 x^{- 4} = \frac{24}{x^{4}}

We must now determine when

\frac{24}{x^{4}}

is positive or negative.
It's necessary to note that

x^{4}

will always be positive, so

\frac{24}{x^{4}}

will also always be positive. Recall that the domain of f excludes 0, so we know that
f is convex on the interval

(- \infty, 0) \cup (0, + \infty)

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