2025 Level 2 Calculus Paper

Step 1

Find the critical points

Set the derivative equal to zero and solve. Type the two x-values separated by a comma.

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Step 2

Differentiate \(f'(x)=2x^2+2x-24\) one more time.

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Step 3

Which critical point gives a local minimum?

Step 4

Integrate the gradient function to get \(f(x)\).

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Step 5

The local minimum happens at \(x=3\), and its \(y\)-value is \(-46\). Which equation should we use?

Step 6

Now solve for \(c\) and type the full function.

f(x)=\frac{2}{3}x^3+x^2-24x+c \] \[ f(3)=\frac{2}{3}(27)+9-72+c=-46 \] \[ 18+9-72+c=-46 \] \[ -45+c=-46 \] \[ c=-1

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