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Find the equation of the tangent line to the graph of the function f(x)=(x 2

+6)(x−4) at the point (1,−21). y= 圕固

User Todd Lyons
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Answer:

To find the equation of the tangent line to the graph of the function \(f(x) = (x^2 + 6)(x - 4)\) at the point \((1, -21)\), you can follow these steps:

1. Find the derivative of \(f(x)\) to get the slope of the tangent line.

2. Use the point-slope form of the equation of a line to find the equation of the tangent line.

Let's start with step 1:

\(f(x) = (x^2 + 6)(x - 4)\)

Use the product rule to find the derivative \(f'(x)\):

\(f'(x) = (x - 4) \cdot \frac{d}{dx}(x^2 + 6) + (x^2 + 6) \cdot \frac{d}{dx}(x - 4)\)

\(f'(x) = (x - 4) \cdot (2x) + (x^2 + 6) \cdot 1\)

\(f'(x) = 2x(x - 4) + (x^2 + 6)\)

Now, let's find the slope of the tangent line at the point \((1, -21)\) by plugging \(x = 1\) into \(f'(x)\):

\(f'(1) = 2(1)(1 - 4) + (1^2 + 6)\)

\(f'(1) = 2(-3) + 7\)

\(f'(1) = -6 + 7\)

\(f'(1) = 1\)

So, the slope of the tangent line at \((1, -21)\) is \(m = 1\).

Now, move on to step 2, where we use the point-slope form of the equation of a line:

\(y - y_1 = m(x - x_1)\)

where \((x_1, y_1)\) is the point \((1, -21)\) and \(m\) is the slope we just found.

Plugging in the values:

\(y - (-21) = 1(x - 1)\)

Simplify:

\(y + 21 = x - 1\)

Now, isolate \(y\):

\(y = x - 1 - 21\)

\(y = x - 22\)

So, the equation of the tangent line to the graph of \(f(x) = (x^2 + 6)(x - 4)\) at the point \((1, -21)\) is \(y = x - 22\).

Explanation:

User Deanie
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