Find f(x) if f(1) = −1 and the tangent line at (x, f(x)) has slope7ex + 6.

F(x)=

if slope is 7ex + 6

then f(x) = (7/2)e x^2 + 6x + c
but (1,-1) lies on it
-1 = (7/2)e (1) + 6 + c
-2 = 7e + 12 + 2c
2c = -14 - 7e
c = -7 - 7e/2

f(x) = (7e/2)x^2 + 6x - 7 - 7e/2

To find the function f(x), we need to integrate the given slope expression. Let's first integrate the expression 7ex + 6 with respect to x:

∫ (7ex + 6) dx

Using the power rule of integration, we can integrate each term separately:

∫ 7ex dx + ∫ 6 dx

∫ 7ex dx = 7 ∫ ex dx = 7ex + C1 (where C1 is the constant of integration)

∫ 6 dx = 6x + C2 (where C2 is another constant of integration)

Now let's combine these two integrals:

∫ (7ex + 6) dx = 7ex + C1 + 6x + C2

Since the derivative of a function gives the slope of its tangent line at any point, we need to find the derivative of f(x) to verify if the tangent line has the desired slope. Taking the derivative of the integrated expression gives us:

f'(x) = d/dx (7ex + C1 + 6x + C2)
= 7ex + 6

Since the given slope of the tangent line is 7ex + 6, and the derivative of f(x) matches that expression, we can conclude that the function f(x) is:

f(x) = 7ex + 6x + C

where C is the constant of integration. However, we still need to find the value of C using the given condition f(1) = -1. Plugging in x = 1 into our function, we get:

f(1) = 7e(1) + 6(1) + C = -1

Simplifying this equation, we get:

7e + 6 + C = -1

C = -7e - 7

Substituting the value of C back into our function, we have:

f(x) = 7ex + 6x - 7e - 7

Therefore, the function f(x) is 7ex + 6x - 7e - 7.

To find the function f(x), we can start by using the given information about the slope of the tangent line. The slope of a tangent line at any point (x, f(x)) on the graph of a function represents the derivative of the function at that point.

In this case, the slope of the tangent line is given as 7ex + 6. Since the slope of the tangent line is the derivative of the function, we can find f'(x) (the derivative of f(x)) by setting it equal to 7ex + 6:

f'(x) = 7ex + 6

Now, we can find f(x) by performing integration on f'(x):

∫ f'(x) dx = ∫ (7ex + 6) dx

Integrating 7ex with respect to x gives us 7e^x, and integrating 6 with respect to x gives us 6x:

f(x) = 7e^x + 6x + C

where C is the constant of integration.

To determine the specific value of the constant of integration, we can use the given information that f(1) = -1. Plug in x = 1 into the equation for f(x) and solve for C:

-1 = 7e^1 + 6(1) + C
-1 = 7e + 6 + C

Simplifying the equation gives:

C = -1 - 7e - 6
C = -7e - 7

Now we have the complete equation for f(x):

f(x) = 7e^x + 6x - 7e - 7