Question to test your Curve Sketching knowledge

Question

Using the first and second derivative, determine the intervals of increase and decrease and the intervals on which the graph of the function is concave up and concave down. Then, sketch the curve defined by the function.

$y = x^{\frac{1}{3}} + {(x - 2)}^{\frac{1}{3}}$

$y^{'} = \frac{1}{3} (\frac{1}{x^{\frac{2}{3}}} + \frac{1}{{(x - 2)}^{\frac{2}{3}}})$

$y^{''} = - \frac{2}{9} (\frac{1}{x^{\frac{5}{3}}} + \frac{1}{{(x - 2)}^{\frac{5}{3}}})$

Accepted Answer

Increasing on $(- \infty, \infty)$

Concave up on $open paren negative infinity comma 0 close paren$ and $open paren 1 comma 2 close paren$

Concave down on $open paren 0 comma 1 close paren$ and $open paren 2 comma infinity close paren$

Answer

Decreasing on $(- \infty, \infty)$

Concave up on $open paren negative infinity comma 0 close paren$ and $open paren 2 comma infinity close paren$

Concave down on $open paren negative infinity comma 0 close paren$ and $open paren 2 comma infinity close paren$

Answer

Decreasing on $(- \infty, \infty)$

Concave up on $open paren negative infinity comma 0 close paren$ and $open paren 0 comma infinity close paren$

Concave down on $(- \infty, - 2)$ and $open paren negative 1 comma 0 close paren$

Answer

Increasing on $(- \infty, \infty)$

Concave up on $open paren negative infinity comma 0 close paren$ and $open paren 0 comma infinity close paren$

Concave down on $(- 2, - 1)$ and $open paren 0 comma infinity close paren$

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