Concavity and Inflection Points – Practice Questions

Below are calculus practice questions with detailed solutions on concavity and inflection points of function graphs.

Question 1

Determine the concavity of the graph of the general quadratic function defined by

Solution

• Compute the derivatives: \[ f'(x) = 2ax + b \] \[ f''(x) = 2a \]
• The sign of \( f''(x) \) depends only on \( a \):
  • If \( a > 0 \), the graph is concave up.
  • If \( a < 0 \), the graph is concave down.

Question 2

Find all intervals where the function

is concave up.

Solution

• \[ f'(x) = \cos x, \qquad f''(x) = -\sin x \]
• Concavity up occurs when \[ f''(x) > 0 \quad \Longrightarrow \quad \sin x < 0 \]
• Over one period \( [0, 2\pi] \), this happens on \( (\pi, 2\pi) \).
• Therefore, \[ (\pi + 2\pi k,\; 2\pi + 2\pi k), \quad k \in \mathbb{Z} \]

Question 3

The graph of \( f'(x) \) is shown below for \( x \in [a,g] \). On which intervals is \( f \) decreasing and concave down? Also find all inflection points.

Solution

• \( f \) is decreasing where \( f'(x) < 0 \).
• \( f \) is concave down where \( f'(x) \) is decreasing.

• Decreasing and concave down on: \[ (0,d) \quad \text{and} \quad (e,g) \]
• Inflection points: \[ (c,f(c)),\ (d,f(d)),\ (e,f(e)),\ (f,f(f)) \]

Question 4

Find all inflection points of

Solution

• \[ f'(x) = 16x^3 - 3x^2 \] \[ f''(x) = 48x^2 - 6x = 6x(8x - 1) \]
• Sign changes occur at: \[ x = 0, \quad x = \frac{1}{8} \]
• Inflection points: \[ (0,2), \quad \left(\frac{1}{8}, \frac{2047}{1024}\right) \]

Question 5

Determine the inflection point of

Then find the inflection point of

Solution

• \[ f'(x) = -3x^2 + 6x, \quad f''(x) = -6x + 6 \]
• \[ f''(x) = 0 \Rightarrow x = 1 \]
• Inflection point of \(f\): \[ (1,3) \]
• Since \( g(x) = f(x-2) \), the graph shifts right 2 units.
• Inflection point of \(g\): \[ (3,3) \]

More Calculus Resources

Calculus questions with solutions
Calculus tutorials and practice problems