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Home Practice Questions AP Calculus AB Analytical Applications of Differentiation Question No. 23

23. The second derivative of the function \( f \) is given by \( f''(x) = x(x+a)(x-e)^2 \) and the graph of \( f'' \) is shown above. For what values of \( x \) does the graph of \( f \) have a point of inflection?






Answer is: option4

a and 0

Solution:

At point a: The graph crosses the x-axis. This suggests a sign change → possible inflection point.

At point 0: The graph crosses the x-axis. Sign change again → possible inflection point.

At point e: The graph touches the x-axis but does not cross it. No sign change → not a point of inflection.

So the points of inflection are where the graph crosses the x-axis: at a and 0.

Final Answer: a and 0.

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