The derivative of the function f is given by f′(x)=−3x+4 for all x, and f(−1)=6. Which of the following is an equation of the line tangent to the graph of f at x=−1 ?
y=7x+13The slope of the line tangent to the graph of ff at x=−1 is f′(−1)=−3(−1)+4=7. An equation of the line containing the point (−1,6) with slope 7 is y=7(x+1)+6=7x+13
To find the equation of the line tangent to the graph of f at x=-1, we need to use the point-slope form of the equation of a line, which is given by:
y – y1 = m(x – x1)
where m is the slope of the line and (x1, y1) is the point of tangency.
We are given that f(-1) = 6, so (x1, y1) = (-1, 6). We also know that the slope of the tangent line at x=-1 is f'(-1) = -3(-1) + 4 = 7.
Therefore, the equation of the tangent line is:
y – 6 = 7(x + 1)
or, equivalently:
y = 7x + 13
Hence, the correct answer is (C) y = 7x + 13.
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