Given:
[tex]f\mleft(x\mright)=ax^3+bx[/tex]It has a local maximum at the point (1,6).
To find:
The constants a and b.
Explanation:
Since (1, 6) is the local maximum of the function.
So, the function must be passing through the point (0, 6).
Substituting x = 1, and y = 6, we get
[tex]\begin{gathered} 6=a(1^3)+b \\ 6=a+b \\ a+b=6..........\left(1\right) \end{gathered}[/tex]Since (1, 6) is the local maximum of the function.
So, x = 1 must be one of its critical points.
Let us find the derivative of the function.
[tex]\begin{gathered} f^{\prime}(x)=3ax^2+b \\ f^{\prime}(1)=0 \\ 3a+b=0.........(2) \end{gathered}[/tex]Subtracting (1) from (2), we get
[tex]\begin{gathered} 2a=-6 \\ a=-3 \end{gathered}[/tex]Substituting a = -3 in the equation (1) we get,
[tex]\begin{gathered} -3+b=6 \\ b=9 \end{gathered}[/tex]Therefore, the constants are
[tex]\begin{gathered} a=-3 \\ b=9 \end{gathered}[/tex]Final answer:
The constants are
[tex]\begin{gathered} a=-3 \\ b=9 \end{gathered}[/tex]