Question

Suppose h(x) = ax^3 + 11x^2. Determine a if h has an inflection point at x = 4. Show all your work.

Answer 1

Answer:

$a=-11/12$

Step-by-step explanation:

We are given the function:

$h(x)=ax^3 + 11 x^ 2$

And we want to determine a such that h(x) has an inflection point at x = 4.

Possible inflection points are whenever h''(x) equals 0. So, we will first differentiate h(x) twice. This yields:

$h'(x)=3ax^2+22x$

So:

$h^{\prime\prime} (x)=6ax+22$

Inflection points occur when h''(x) = 0. So:

$0=6ax+22$

Since we have an inflection at x = 4:

$0=24a+22$

And solving for a yields:

$a=-22/24=-11/12$