Answer :
To find f′(x) and the equation of the tangent line to the graph of f at x=2, let's follow these steps Find f′(x) by differentiating f(x) with respect to x.
To differentiate f(x)=x(4−x)^3, we can use the product rule. The product rule states that if we have two functions u(x) and v(x), the derivative of their product is given by (u(x)v'(x) + v(x)u'(x)).
Let's apply the product rule to f(x)=x(4−x)^3:
f(x) = x * (4−x)^3
Using the product rule, we differentiate f(x) as follows:
f′(x) = x * [3(4−x)^2 * (-1) + (4−x)^3 * 1]
Simplifying this expression gives:
f′(x) = x * [-3(4−x)^2 + (4−x)^3]
So, the equation of the tangent line to the graph of f at x=2 is:
y - 16 = -8(x - 2)
This equation represents the line that is tangent to the graph of f at x=2, with a slope of -8.
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