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# Draw sketches of possible graphs for which the following data hold. Consider only the domain $0 \leqslant x \leqslant 5$, and assume that the graph of $y=f^{\prime \prime}(x)$ is smooth. (a) f(0) = 0, f(2) = 5, f'(2) = 0, f"(2) < 0, f(4) = 3, f'(4) = 0, f"(4) > 0 (b) f(0) = 0, f"(1) < 0, f(2) = 5, f'(2) = 0, f"(3) > 0, f(4) = 7 (c) f(0) = 0, f'(0) = -1, f'(1) = 0, f(3) = 0, f'(3) = 2, f"(3) = 0, f"(4) < 0 (d) f(0) = 1, f'(0) = 1, f''(0) = 1 and f"(x) increases as x increases (e) f(0) = 1, f'(0) = 0, f'(x) < 0 for 0 < x < 5, f(5) = f'(5) = 0 (f) f(0) = 3, f'(0) = -2, f"(x) > 0 for 0 < x < 5, f(5) = f'(5) = 0

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Before sketching a possible graphs with given conditions about derivative requirements, we need to know that:

If $f'\left(x \right)=0$, then $f\left(x \right)$ has maximum or minimum and the slope is $m=0$.

If $f'\left(x \right)<0$, then $f\left(x \right)$ is decreasing.

If $f'\left(x \right)>0$, then $f\left(x \right)$ is increasing.

If $f''\left(x \right)<0$, then $f\left(x \right)$ is concave down ($\cap$).

If $f''\left(x \right)>0$, then $f\left(x \right)$ is concave up ($\cup$).

If $f''\left(x \right)=0$, then $x$ is the inflection point, i.e. function changes from concave down to concave up and vice-versa.

Further, if there is given condition as $f\left(x \right)=y$, then we will draw the point $\left(x,y\right)$ and through that point function must pass.

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