It is given that at $x = 1$, the function $f(x) = x^4 - 62x^2 + ax + 9$ attains its maximum value on the interval $[0, 2]$.
Find the value of $a$.

Q: It is given that at $x = 1$, the function $f(x) = x^4 - 62x^2 + ax + 9$ attains its maximum value on the interval $[0, 2]$. Find the value of $a$.

Explanation: We are given: \[ f(x) = x^4 - 62x^2 + ax + 9 \] Differentiate: \[ f'(x) = 4x^3 - 124x + a \] Since $f(x)$ has a maximum at $x = 1$, we have: \[ f'(1) = 0 \] Substitute $x = 1$: \[ f'(1) = 4(1)^3 - 124(1) + a = 4 - 124 + a = 0 \] Solve for $a$: \[ a = 120 \] \[ \boxed{a = 120} \]

Question

It is given that at $x = 1$, the function $f(x) = x^4 - 62x^2 + ax + 9$ attains its maximum value on the interval $[0, 2]$.  
Find the value of $a$.

Accepted Answer

Explanation: We are given:
$$
f(x) = x^4 - 62x^2 + ax + 9
$$

Differentiate:
$$
f'(x) = 4x^3 - 124x + a
$$

Since $f(x)$ has a maximum at $x = 1$, we have:
$$
f'(1) = 0
$$

Substitute $x = 1$:
$$
f'(1) = 4(1)^3 - 124(1) + a = 4 - 124 + a = 0
$$

Solve for $a$:
$$
a = 120
$$

$$
\boxed{a = 120}
$$

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