Practice Curve analysis
Curve analysis combines the domain, roots, derivatives and limiting behaviour to describe a function.
Use these properties together to justify the shape of its graph.
Exercises with solutions
In a curve sketch you examine a function in a fixed order: domain, zeros, derivatives, extrema, inflection points and behaviour as x tends to infinity. Extrema come from the zeros of the first derivative; the second derivative decides between maximum and minimum.
Exercise 1
Find the zeros of f(x) = x² − 4x.
Show solution and working
x = 0 and x = 4
Factor out x: x · (x − 4) = 0, so x = 0 or x = 4.
Exercise 2
Find the extremum of f(x) = x² − 4x.
Show solution and working
x = 2, a minimum
f′(x) = 2x − 4. Setting it to zero gives x = 2. Since f″(x) = 2 > 0 this is a minimum.
Exercise 3
How do you find an inflection point?
Show solution and working
from the zero of the second derivative
At an inflection point the curvature changes. There f″(x) = 0.
Exercise 4
How does f(x) = −x³ behave for large x?
Show solution and working
it tends to −infinity
The highest exponent is 3 and therefore odd; the factor −1 makes the behaviour for large x negative.
Common mistake
Checking only the first derivative and not determining the type of extremum. Only the second derivative or a sign change decides whether it is a maximum or a minimum.
Keep learning with MatheJump
Create a free account and explore math practice. Choose from the available exercises to suit your learning needs.