Question

Estimate graphically the local maximum and local minimum of f(x) = 4x2 + 3x + 2

Answer 1

To estimate the local maximum and local minimum of the function f(x) = 4x^2 + 3x + 2, we can find the vertex of the parabolic graph. The vertex is (-3/8, 23/16), which represents the local minimum. There is no local maximum for this function.

Step-by-step explanation:

To graph the function f(x) = 4x^2 + 3x + 2 and estimate the local maximum and local minimum, we can start by finding the vertex of the parabolic function. The x-coordinate of the vertex can be found using the formula x = -b/(2a), where a, b, and c are the coefficients of the quadratic equation f(x). In this case, a = 4 and b = 3. Plugging these values into the formula, we get x = -3/(2*4) = -3/8. To find the y-coordinate, substitute the x-coordinate into the equation f(x):

f(-3/8) = 4(-3/8)^2 + 3(-3/8) + 2 = 4(9/64) - 9/8 + 2 = 9/16 - 9/8 + 2 = 9/16 - 18/16 + 32/16 = 23/16.

Therefore, the vertex of the parabola is (-3/8, 23/16), and this point represents the local minimum since the coefficient of x^2 is positive. We can estimate the local maximum by examining the behavior of the graph as x approaches -∞ and +∞. As x becomes very large (positive or negative), the quadratic function will tend towards positive infinity. Therefore, there is no local maximum for this function.

Answer 2

Seeing the graph from desmos.com, we know there is only one place where the graph will go up then down, or down then up given that the maximum exponent of x is it squared. That place, as seen, is -0.375 for the minimum as it goes down then up. There's no maximum in sight :(