Final answer:
To find the second derivative of y=ax³, differentiate y'=3ax² with respect to x, substitute a=7 into the second derivative, and evaluate y'' at x=3. The value of y'' at x=3 is 126.
Step-by-step explanation:
To find the second derivative of y=ax³, we first need to find the first derivative of y. The first derivative of y=ax³ is y'=3ax². Now, to find the second derivative, we differentiate y'=3ax² with respect to x. Differentiating 3ax² gives us y''=6ax.
Given that a=7, we substitute a=7 into y''=6ax to find y''=6(7)x=42x.
Now we can find y'' at x=3 by substituting x=3 into y''=42x. Therefore, y'' at x=3 is 42(3)=126.