# Exam-Style Question on Graphs

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

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Question id: 602. This question is similar to one that appeared on an IB AA Standard paper in 2021. The use of a calculator is not allowed.

The following diagram shows the graph of $$f'$$, the first derivative of a function $$f$$.

The graph of $$f'$$ has x-intercepts at $$x=a, x=c, x=e \text{ and } x=g$$. It has local maximum points at $$x=b \text{ and } x=f$$ and a local minimum point at $$x=d$$.

(a) Find all the values of $$x$$ where the graph of $$f$$ is increasing. Justify your answer.

(b) Find all the values of $$x$$ where the graph of $$f$$ has a local maximum. Justify your answer.

(c) Find all the values of $$x$$ where the graph of $$f$$ has a local minimum. Justify your answer.

(d) Find all the values of $$x$$ where the graph of $$f$$ has points of inflection and determine which of these is a horizontal point of inflection.

(e) The total area of the region enclosed by graph of $$f'$$ and the x-axis for $$a \lt x \lt e$$ is 6.

Given that $$f(a) + f(e) = 3$$, find the value of $$f(c)$$.

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