- Topic
- Curve sketching
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Curve sketching practice question 15
The bell-shaped curve has two inflection points, placed symmetrically about the -axis. Find the -coordinate of the one with .
The curve bends one way near its peak and the other way in its tails. Which derivative changes sign where the bending switches?
Inflection points are where changes sign. Start from .
Differentiate with the product rule, factor out , and find where what remains changes sign.
A whole number.
Theory UBC Math 100 §7CLP-1 §3.6Key results
Worked solution
Strategy
Differentiate twice, factor out the exponential, which is always positive, and read the sign of what remains.
Working
Chain rule: the derivative of −x²/2 is −x.
Product rule on −x times e−x²/2, then factor.
e−x²/2 > 0, so f″ has the sign of x² − 1, which changes at x = ±1.
Concave down around the peak, concave up in the two tails; the positive one is x = 1.
Answer
The answer is 1.
Check
The bell is steepest at x = 1, with slope −0.607 there against −0.600 at x = 0.9 and −0.601 at x = 1.1, as an inflection point needs.
Where this goes wrong
Solving f′ = 0 instead of f″ = 0, which gives x = 0: the peak, where the curve is concave down, not a change of concavity. Dropping the first product-rule term of f″ points to x = 0 as well.
Leans on
The product rule, from Differentiation rules
Try a similar question
Find the -coordinate of the inflection point of .Saved in this browser only.