The function \( f(x) = x^3 - 6x^2 + 9x + c \) has a local maximum at \( x = 1 \). Find \( c \) if \( f(1) = 2 \).
["Title: How to Find ( c ) in ( f(x) = x^3 - 6x^2 + 9x + c ) When It Has a Local Maximum at ( x = 1 ) and ( f(1) = 2 )", "---", "### Understanding Local Maxima and How to Use Them in Polynomial Functions", "When analyzing functions like ( f(x) = x^3 - 6x^2 + 9x + c ), identifying critical points—especially local maxima—is essential for graphing and optimization. One powerful technique involves using derivatives to determine where such extrema occur. In this article, we explore why ( f(x) = x^3 - 6x^2 + 9x + c ) has a local maximum at ( x = 1 ) and how to find the constant ( c ) when ( f(1) = 2 ).", "---", "### Step 1: Find the First Derivative", "To locate local maxima, start by finding the first derivative ( f'(x) ), which gives the slope of the tangent line at any point.", "Given:\n[\nf(x) = x^3 - 6x^2 + 9x + c\n]", "Differentiate term by term:\n[\nf'(x) = 3x^2 - 12x + 9\n]", "---", "### Step 2: Confirm a Critical Point at ( x = 1 )", "A local maximum occurs where the first derivative is zero and changes sign from positive to negative. Check ( f'(1) ):\n[\nf'(1) = 3(1)^2 - 12(1) + 9 = 3 - 12 + 9 = 0\n]", "Since ( f'(1) = 0 ), ( x = 1 ) is a critical point.", "---", "### Step 3: Confirm It’s a Local Maximum Using the Second Derivative", "To confirm ( x = 1 ) is a local maximum, compute the second derivative:\n[\nf''(x) = \frac{d}{dx}(3x^2 - 12x + 9) = 6x - 12\n]", "Evaluate at ( x = 1 ):\n[\nf''(1) = 6(1) - 12 = -6 < 0\n]", "Since ( f''(1) < 0 ), the function has a local maximum at ( x = 1 )—confirming our earlier result.", "---", "### Step 4: Use ( f(1) = 2 ) to Solve for ( c )", "We are given that ( f(1) = 2 ). Plug ( x = 1 ) into the original function:\n[\nf(1) = (1)^3 - 6(1)^2 + 9(1) + c = 1 - 6 + 9 + c = 4 + c\n]", "Set equal to 2:\n[\n4 + c = 2 \Rightarrow c = 2 - 4 = -2\n]", "---", "### Conclusion", "For the cubic function ( f(x) = x^3 - 6x^2 + 9x + c ) to have a local maximum at ( x = 1 ), the value of ( c ) must be adjusted so that ( f(1) = 2 ). Through derivative analysis—verifying ( f'(1) = 0 ) and ( f''(1) < 0 )—we confirm a local maximum exists at ( x = 1 ). Solving ( f(1) = 2 ) gives ( c = -2 ).", "Key Takeaway:\nLocal maxima in polynomial functions can be confirmed via critical points and second derivative tests. While ( c ) doesn’t affect the location of critical points, it sets the vertical position—making it crucial when a specific function value like ( f(1) = 2 ) is given.", "---", "Keywords:\n( f(x) = x^3 - 6x^2 + 9x + c ), local maximum, ( x = 1 ), derivative, second derivative test, find ( c ), polynomial functions, critical point, optimization, calculus, youtube math tutorial, real functions.", "Meta Description:\nExplore how to determine the constant ( c ) in ( f(x) = x^3 - 6x^2 + 9x + c ) when it has a local maximum at ( x = 1 ), using derivatives. Learn how ( f(1) = 2 ) leads to ( c = -2 ). Perfect for students and math learners."]
