f(x)=ax2+7x+c𝑓(𝑥)=𝑎𝑥2+7𝑥+𝑐In the given quadratic function, a𝑎 and c𝑐 are constants. The graph of y=f(x)𝑦=𝑓(𝑥) in the xy𝑥𝑦-plane is a parabola that opens downwards and has a vertex at the point (h,k)(ℎ,𝑘), where hℎ and k𝑘 are constants. If k>0𝑘>0 and f(−3)=f(4)𝑓(-3)=𝑓(4), which of the following must be true?I. c>0𝑐>0II. a≤−1

Given the quadratic function f(x) = ax^2 + 7x + c, we know that the graph of this function is a parabola that opens downwards. This means that the coefficient a of the x^2 term must be negative.

The vertex of the parabola is given by the point (h, k), and we know that k > 0. This means that the y-coordinate of the vertex is positive, which implies that the parabola opens downwards and its maximum point is above the x-axis.

We are also given that f(-3) = f(4). This means that the y-values of the function at x = -3 and x = 4 are the same. This implies that the axis of symmetry of the parabola is midway between -3 and 4, which is at x = 0.5.

Now, let's analyze the statements:

I. c > 0: The constant c is the y-intercept of the parabola. Since the parabola opens downwards and its maximum point is above the x-axis, the y-intercept must also be above the x-axis. Therefore, c must be greater than 0.

II. a ≤ -1: As we mentioned before, the coefficient a must be negative because the parabola opens downwards. However, we cannot determine a specific value or range for a based on the given information. Therefore, we cannot say that a is necessarily less than or equal to -1.

So, the statement I must be true, while the statement II may or may not be true.