Final answer:
The inflection points for the density function of the standard normal random variable X are at x = -1 and x = 1. These are one standard deviation from the mean in both directions for the standard normal distribution, which has a mean of 0 and a standard deviation of 1.
Step-by-step explanation:
You're asking about the inflection points of the density function for a normal random variable X. An inflection point in the context of the normal distribution, which is a continuous probability distribution, is where the second derivative of the probability density function changes sign. For the standard normal distribution, which is symmetric and has a bell curve shape, the inflection points occur where the curvature changes from concave to convex or vice versa.
For a standard normal distribution, which is denoted as X~N(0,1), the inflection points are always one standard deviation from the mean. This is because the probability density function of the normal distribution, which is symmetric about the mean, has its maximum curvature at these points. In this case, the mean (μ) is 0 and the standard deviation (σ) is 1, which means the inflection points are at μ - σ and μ + σ, or x = -1 and x = 1. Therefore, the values of x that are inflection points for the density function of the normal random variable X are x = -1 and x = 1. This means the correct option in the final answer is B) x=0 C) x=1.