Find constants a,b, and c such that the function f(x)=ax^3+bx^2+c will have a local extremum at (2,11)and a point of inflection at (1,5). Sketch the graph of y=f(x).
How would I solve this.
Thank you.
To find the constants a, b, and c that satisfy the given conditions, we can follow these steps:
Step 1: Find the first derivative of the function f(x) with respect to x. The first derivative represents the slope of the function and is denoted as f'(x).
f'(x) = 3ax^2 + 2bx
Step 2: Since we want a local extremum at (2,11), we know that the slope of the tangent line at x = 2 should be zero. This means that f'(2) = 0.
Substitute 2 for x in f'(x) and set it equal to zero:
f'(2) = 3a(2)^2 + 2b(2) = 0
Simplify the equation: 12a + 4b = 0 -- (Equation 1)
Step 3: Similarly, since we want a point of inflection at (1,5), we know that the second derivative of f(x) should be zero at x = 1. The second derivative is denoted as f''(x).
f''(x) = 6ax + 2b
Step 4: Substitute 1 for x in f''(x) and set it equal to zero:
f''(1) = 6a(1) + 2b = 0
Simplify the equation: 6a + 2b = 0 -- (Equation 2)
Step 5: Solve the system of equations formed by (1) and (2) to find the values of a and b.
Multiplying Equation 2 by 2 gives us: 12a + 4b = 0.
Comparing this with Equation 1, we can see that the two equations are identical. This indicates that there is no unique solution for a and b. Therefore, we can assume any value for a or b, provided they satisfy Equation 1 and Equation 2.
For simplicity, let's assume a = 1. Substituting this value into Equation 1:
12(1) + 4b = 0
12 + 4b = 0
4b = -12
b = -3
So, we have a = 1 and b = -3.
Step 6: To find the value of c, substitute the values of a and b into the original function f(x) and solve for c:
f(x) = ax^3 + bx^2 + c
f(x) = x^3 - 3x^2 + c
Since f(x) has a local extremum at (2,11), substitute x = 2 and y = 11 into the equation:
11 = 2^3 - 3(2)^2 + c
11 = 8 - 12 + c
11 - 8 + 12 = c
c = 15
Therefore, the constants that satisfy the given conditions are a = 1, b = -3, and c = 15.
To sketch the graph y = f(x), use the obtained values of a, b, and c and plot the points (2,11) and (1,5) on the graph.