Video Transcript
If 𝑦 equals two 𝑥 to the fourth power minus four 𝑥 cubed minus 24𝑥 plus 50, for which of the following values of 𝑥 does the graph of 𝑦 have a point of inflection? a) Negative three. b) Two. c) One. Or d) Negative two.
Let’s remind ourselves first then what is meant by a point of inflection. This is a point on the graph of a function where its concavity changes, either from concave downward to concave upward or vice versa. When a graph is concave downward, its slope or first derivative will be decreasing. And therefore, its second derivative d two 𝑦 by d𝑥 squared will be negative. Whereas when a graph is concave upward, its slope or first derivative is increasing and hence the second derivative will be positive. At a point inflection itself, the second derivative of the function will be equal to zero. And whilst this is a necessary condition for a point of inflection, it isn’t sufficient because it’s also possible for the second derivative of a function to be equal to zero as a local minima or the local maxima.
In order to ensure that the point is a point of inflection, the second derivative must not only be equal to zero but must also undergo a change of sign. So to determine at what 𝑥 values our function has point of inflection, we need to find its first and then its second derivative which we can do using differentiation. By the power rule of differentiation, the first derivative d𝑦 by d𝑥 is equal to two multiplied by four 𝑥 cubed minus four multiplied by three 𝑥 squared minus 24. Remember, the derivative of a constant — in this case, positive 50 — is just zero.
The first derivative simplifies to eight 𝑥 cubed minus 12𝑥 squared minus 24. To find the second derivative, we differentiate again, giving eight multiplied by three 𝑥 squared minus 12 multiplied by two 𝑥, which simplifies to 24𝑥 squared minus 24𝑥. Now remember at point of inflection, the second derivative is equal to zero. So we’re going to set this expression equal to zero and solve for 𝑥. We have 24𝑥 squared minus 24𝑥 equals zero, a quadratic equation in 𝑥. But it’s a quadratic that can be solved by factoring. We take a common factor of 24𝑥 from each of our terms, giving 24𝑥 multiplied by 𝑥 minus one is equal to zero.
To solve, we take each of our factors in turn, set them equal to zero, and solve the resulting equations. 24𝑥 equals zero leads to 𝑥 equals zero and 𝑥 minus one equals zero leads to 𝑥 equals one. So we find that there are two possible 𝑥-values at which this function has points of inflection. Remember, though, that the second derivative being equal to zero is not sufficient to be able to conclude that the graph does have a point of inflection, as it could also be a local minimum or a local maximum. We, therefore, need to check for a change in sign around each of these 𝑥 values.
We’ll, now evaluate the second derivative of our function at 𝑥-values a little either side of these two values of zero and one. So I’ve chosen the values of negative 0.5, 0.5, and 1.5. It’s perhaps a little bit easier to work with fractions rather than decimals as we don’t have a calculator. So substituting negative 0.5 into our second derivative first of all, gives 24 multiplied by negative one-half squared minus 24 multiplied by negative one-half. That’s 24 multiplied by one-quarter plus 24 multiplied by one-half or six plus 12 which is equal to 18. In the same way, we can evaluate the second derivative when 𝑥 is equal to 0.5, giving negative six, and when 𝑥 is equal to 1.5, giving 18.
So looking either side of the possible point inflection when 𝑥 is equal to zero, we see that the second derivative does indeed undergo a change of sign from positive to negative. Looking around the possible point of inflection when 𝑥 is equal to one, we see that the second derivative also undergoes a change of sign here, from negative to positive. And therefore, this function has points of inflection at both 𝑥 equals zero and 𝑥 equals one. However, 𝑥 equals zero isn’t included in the four options we were given. So our answer to the question is option c). The graph of 𝑦 equals two 𝑥 to the fourth power minus four 𝑥 cubed minus 24𝑥 plus 50 has a point of inflection when 𝑥 is equal to one.
