Consider a function \(f\) satisfying the following conditions.

  • \(f(0)=0\) and \(f'(0)=0\).
  • \(\lim _{x\to -5}f(x) = -\infty \).
  • \(f'(x)<0\) on \((-\infty ,-5)\) and \((0,\infty )\).
  • \(f'(x)>0\) on \((-5,0)\).
  • \(f''(x)<0\) on \((-15,-5)\) and \((-5,15)\).
  • \(f''(x)>0\) on \((-\infty ,-15)\) and \((15,\infty )\).

\(f\) is increasing and concave down on the interval \(\left (\answer {-5},\answer {0}\right )\)

\(f\) is increasing and concave up on NO interval.

\(f\) is decreasing and concave down on the intervals (from left to right) \(\left (\answer {-15},\answer {-5}\right )\) and \(\left (\answer {0},\answer {15}\right )\).

\(f\) is decreasing and concave up on the intervals (from left to right) \(\left (\answer {-\infty },\answer {-15}\right )\) and \(\left (\answer {15},\answer {\infty }\right )\)

\(f\) has \(\answer {0}\) local minima.

\(f\) has a local maximum at \(x=\answer {0}\).

\(f\) has inflection points (from left to right) at \(x=\answer {-15}\) and \(x=\answer {15}\).

[Picture][Picture]

[Picture][Picture]

Which diagram shows the graph of a function that has all of the above properties?

\(A\) \(B\) \(C\) \(D\)