# Based on the function graph, in how many points the derivative equals 2?

I need to answer the question in the title for this function graph.

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I see that the derivative is positive in $3$ segments of the graph, and thinking about it as roughly $\frac{\bigtriangleup y} {\bigtriangleup x}$ it should be have points where derivative is $2$ or more in the first $2$ segments where the function grows.

But I don't know where to go from here and how to find the exact number of points.

• It looks like the derivative is greater than 2 at an infinite number of points. – Carser May 9 '16 at 11:43
• @Jed maybe but the question is about points where it equals 2. Also, the answer is supposed to be an integer – dvornik May 9 '16 at 11:49

Hint $1:$ The tangent at any point on the graph gives the derivative at that point.
Hint $2:$ The derivative is zero at exactly $5$ points in the given range.
Hint $3:$ Since the function is continuous, by Intermediate Value Theorem, there must be some $c$ between $a$ and $b$, such that $f(c)=\frac{f(b)-f(a)}{b-a}$
Can you take it from here? (Answer is $2$)