# Exam-Style Question on Differentiation

## A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 22. This question is similar to one that appeared on an IB Studies paper in 2012. The use of a calculator is allowed.

Consider the function $$f(x)=x^3-9x+2$$.

(a) Sketch the graph of $$y=f(x)$$ for $$-4\le x\le 4$$ and $$-14\le y\le 14$$ showing clearly the axes intercepts and local maximum and minimum points. Use a scale of 2 cm to represent 1 unit on the x-axis, and a scale of 1 cm to represent 2 units on the y-axis.

(b) Find the value of $$f(-1)$$.

(c) Write down the coordinates of the y-intercept of the graph of $$f(x)$$.

(d) Find $$f'(x)$$.

(e) Find $$f'(-1)$$

(f) Explain what $$f'(-1)$$ represents.

(g) Find the equation of the tangent to the graph of $$f(x)$$ at the point where x is –1.

R and S are points on the curve such that the tangents to the curve at these points are horizontal. The x-coordinate of R is $$a$$ , and the x-coordinate of S is $$b$$ , $$b \gt a$$.

(h) Write down the value of $$a$$ ;

(i) Write down the value of $$b$$.

(j) Describe the behaviour of $$f(x)$$ for $$a \lt x \lt b$$.

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