Find a function that satisfies the conditions $f ^ { \prime \prime } ( x ) = x ^ { 5 } , f ^ { \prime } ( 0 ) = 9 , f ( 0 ) = 5$ .
A) $f ( x ) = \frac { 1 } { 6 } x ^ { 6 } + 9 x$
B) $f ( x ) = \frac { 1 } { 42 } x ^ { 7 } + 9 x + 5$
C) $f ( x ) = \frac { 1 } { 42 } x ^ { 6 } + 9 x + 5$
D) $f ( x ) = \frac { 1 } { 7 } x ^ { 7 } + 9 x$
E) $f ( x ) = \frac { 1 } { 42 } x ^ { 6 } + 5 x + 9$

Question

Quizplus · Accepted Answer

To find the function $f(x)$ that satisfies the given conditions, we integrate the given second derivative $f''(x) = x^5$ twice. The first integration gives us the first derivative $f'(x)$, and the second integration gives us $f(x)$.1. Integrate $f''(x) = x^5$ to get $f'(x)$:$$f'(x) = \frac{1}{6}x^6 + C_1$$Given $f'(0) = 9$, we find $C_1 = 9$.2. Integrate $f'(x) = \frac{1}{6}x^6 + 9$ to get $f(x)$:$$f(x) = \frac{1}{42}x^7 + 9x + C_2$$Given $f(0) = 5$, we find $C_2 = 5$.Therefore, the function that satisfies all given conditions is $f(x) = \frac{1}{42}x^7 + 9x + 5$, which is option B.

Find a Function That Satisfies the Conditions f′′(x)=x5,f′(0)=9,f(0)=5f ^ { \prime \prime } ( x ) = x ^ { 5 } , f ^ { \prime } ( 0 ) = 9 , f ( 0 ) = 5f′′(x)=x5,f′(0)=9,f(0)=5

Find a Function That Satisfies the Conditions $f ^ { \prime \prime } ( x ) = x ^ { 5 } , f ^ { \prime } ( 0 ) = 9 , f ( 0 ) = 5$