Suppose that f is a function given as f(x) = x^3 + bx^2 + cx...

Suppose that $f$ is a function given as $f(x) = x^3 + bx^2 + cx + 2$. Find the exact values of $b$ and $c$ if we know that the line $y = 29x + 65$ is a tangent line to the graph of $f$ at $x = -3$. b = and c =

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00:01 So in this question, they say, suppose i have f, f is a function given as f of x equals x cubed plus bx squared plus cx plus 5.

00:11 I want to find the exact values of b and c.

00:14 If i know that the line y equals 2x plus 5 is a tangent line to the graph of f at x equals 1.

00:23 So if i know that y equals 2x plus 5 is a tangent line to the graph of f at x equals 1, i know that the value of the function and the tangent line should match up at the point of tangency at x equals 1.

00:41 In other words, f of 1 is the same y value as i get on my tangent line, which is 7.

00:50 So i know that f of 1 equals 7.

00:56 So now, what does this tell you about f of 1? well, if i plug in, i'm getting 1 plus b plus c plus 5, and that's equal to 7.

01:10 So 7 equals 6 plus b plus c.

01:18 I'm getting 1 equals b plus c.

01:21 Okay.

01:23 Now, what else do we know? y equals 2x plus 5 is my tangent line.

01:31 So the slope of my tangeline was 2.

01:34 In other words, f prime of 1 was equal to 2.

01:44 What's my derivative my f prime of x? well, it's 3x squared plus 2bx plus c.

01:56 Now, if i plug 1 into that, my f prime of 1 is 3 plus 2b plus c.

02:04 And that's supposed to be equal to 2, we said...

Question

Suppose that $f$ is a function given as $f(x) = x^3 + bx^2 + cx + 2$. Find the exact values of $b$ and $c$ if we know that the line $y = 29x + 65$ is a tangent line to the graph of $f$ at $x = -3$. b = and c =

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