Topic 5.3: Increasing, Decreasing and Stationary Points Questions

The graph of f(x) has a local minimum at x = 4. What must be true?

Find f′(x) for f(x) = 4x³ − 3x + 7.

What is d/dx[−9]?

The graph of g(x) rises from x = 0 to x = 5, then falls for x > 5. State the sign of g′(x) for (a) 0 < x < 5 and (b) x > 5. State the value of g′(5).

What is the derivative of 8x⁵?

Find dy/dx for y = 3x⁴ − 5x² + 2x − 8.

A function's graph is falling steeply at x = −2. Which is correct?

The function f(x) = x³ − 2x. Which statement about f′(x) is correct?

State the difference between f(a) and f′(a). Give a real-world example of a function and explain what both values represent in that context.

h(x) = −x³ + 6x².

T(x) is the temperature (°C) of a reaction at time x (minutes). The graph of T(x) is a smooth curve that peaks at x = 8. State two things that can be concluded about T′(8).

P(t) is the profit (thousands of dollars) after t years. P′(3) = 4. What does this mean?

h(t) = −5t² + 20t is the height (metres) of a ball at time t (seconds). Without calculating, state what h′(t) = 0 tells you about the motion of the ball.

The graph of f(x) has f(2) = 7 and f′(2) = −3. Describe what the curve looks like at x = 2.

The height h (metres) of a drone at time t (seconds) is shown as a smooth curve. At t = 2 the drone is rising at 3 m/s. At t = 5 the drone reaches its maximum height of 18 m. At t = 8 the drone is falling at 2 m/s. Write the value or sign of h′(t) for each of these three moments, and explain your reasoning.

Find the gradient of y = x² − 5x at x = 3.

The graph of a function passes through A(1, 4) and B(3, 4) with a smooth peak between them. A student says: "f′(1) = f′(3) because the function has the same y-value at both points." Is the student correct? Explain using the meaning of f′.

Differentiate y = x(3x − 4).

Find the gradient of f(x) = 2x³ − 6x at x = −1.

Expand and then differentiate y = (x + 2)(x − 3).