for ax^2 + bx + c with gradient 4x + 2 and a stationary value of 1 find a b and c and nature of point
i think that its a minima
and that its 2x^2 + 2x +2 but im not sure can someone check this please
To find the values of a, b, and c for the quadratic equation ax^2 + bx + c, we can start by looking at the given information.
Given:
1. Gradient of the quadratic is 4x + 2.
2. A stationary value of 1, which implies that the quadratic has a minimum point.
To find the values of a, b, and c, we need to differentiate the quadratic equation and equate it to the given gradient.
Differentiating ax^2 + bx + c with respect to x:
d/dx (ax^2 + bx + c) = 2ax + b
Equating it to the given gradient, 4x + 2:
2ax + b = 4x + 2
By comparing the coefficients of x on both sides, we can determine the values of a and b:
2a = 4 => a = 2
b = 2
Now we can substitute the values of a and b back into the original quadratic equation to find c.
Using a = 2 and b = 2, we have:
ax^2 + bx + c = 2x^2 + 2x + c
Given that the quadratic has a stationary value of 1, we substitute x = 0 into the quadratic equation and set it equal to 1:
2(0)^2 + 2(0) + c = 1
0 + 0 + c = 1
c = 1
Therefore, the values of a, b, and c for the quadratic equation are:
a = 2
b = 2
c = 1
Now, checking the final expression 2x^2 + 2x + 1:
Based on the leading coefficient (a = 2), we see that it is positive, which means the quadratic opens upwards. This confirms that it is indeed a minimum point.
Hence, your expression 2x^2 + 2x + 1 is correct, and the nature of the point is a minimum.