009B Sample Midterm 3, Problem 3 Detailed Solution

Find a curve  $y=f(x)$  with the following properties:

(i)   $f''(x)=6x$

(ii)   Its graph passes through the point  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (0,1)}   and has a horizontal tangent there.

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Background Information:
1. If the graph of  $f(x)$  passes through the point  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (a,b),}   then  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(a)=b.}
2. If  $f(x)$  has a horizontal tangent at the point  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (a,b),}   then  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(a)=0.}

Solution:

Step 1:
Since  $f(x)$  passes through the point  $(0,1),$
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(0)=1.}
Since  $f(x)$  has a horizontal tangent at  $(0,1),$
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(0)=0.}

Step 2:
Now, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {\int f''(x)~dx}\\&&\\&=&\displaystyle {\int 6x~dx}\\&&\\&=&\displaystyle {3x^{2}+C.}\\\end{array}}}
Since  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(0)=0,}   we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {0}&=&\displaystyle {f'(0)}\\&&\\&=&\displaystyle {3(0)^{2}+C}\\&&\\&=&\displaystyle {C.}\\\end{array}}}
Hence,
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x)=3x^{2}.}

Step 3:
We have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {f(x)}&=&\displaystyle {\int f'(x)~dx}\\&&\\&=&\displaystyle {\int 3x^{2}~dx}\\&&\\&=&\displaystyle {x^{3}+D.}\\\end{array}}}
Since  $f(0)=1,$  we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {1}&=&\displaystyle {f(0)}\\&&\\&=&\displaystyle {(0)^{3}+D}\\&&\\&=&\displaystyle {D.}\\\end{array}}}
Hence,
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=x^{3}+1.}

Final Answer:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^3+1}

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