Provide an appropriate response.The graph of the function y = f(x) = x2/3 is shown below. The "V"-shaped graph comes to a sharp point at  Without doing any calculations, decide whether the function has a tangent at   Give reasons for your answer. [Hint: consider the signs of the tangent lines on either side of  and what implication this has in terms of the limit definition of the slope of a tangent line.].

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What will be an ideal response?

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The function does not have a tangent at x = 0. The tangents to the left of x = 0 all have negative slopes, whereas those to the right of x = 0 all have positive slopes. Thus, the limit of the slope as x approaches 0 from the left cannot equal the limit of the slope as x approaches 0 from the right and therefore, according to the definition of the slope of the tangent line, no tangent line exists.

Mathematics