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let f(x)=x^3+px^2+qx,where p and q are constatbts

find the value of p and q so that f(-1)=-8 and f'(-1)=12

By finding the derivative, then putting -1 in the function and the derivative will allow us to calculate the answer.

Since f(x) = x^3 + px^2 + qx, we know that f'(x) = 3x^2 + 2px + q.

From here, we can see that f(-1) = -8 gives

-1^3 + p*(-1)^2 + q(-1) = -8, which translates to

= -1 + p - q = -8. Adding 1 to both sides gives [1] p - q = -7.

Since f'(-1) = 3(-1)^2 + 2p(-1) + q = 12, this reduces to

3 - 2p + q = 12. Subtracting 3 from both sides gives [2] -2p + q = 9.

Adding p-q=-7 and -2p+q=9 together gives -p = 2.

Multiplying both sides by -1 gives p = -2.

Using [1] gives -2 - q = -7. Add 2 to both sides gives -q = -5.

Multiplying by -1 gives q = 5.

Using the 2nd equation to check with p=-2 and q=5, we get 4 + 5 = 9,

and that is correct.

Calculus

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