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

Question

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.

Answer 1

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 $2$)

Answer 2

Hint: The definition of the derivate is

A formula for the coefficient of the tangent.