How do you find the maximum or minimum of quadratic functions?

Answer 1

Please see below.

For a quadratic function #y=ax^2+bx+c#, a maximum is there if #a<0# and it has a minimum, if #a>0#. Please see below for details.
We can write #y=ax^2+bx+c# as
#y=a(x^2+b/ax)+c#
= #a(x^2+2xxb/(2a)xx x+(b/(2a))^2-(b/(2a))^2)+c#
= #a(x^2+2xxb/(2a)xx x+(b/(2a))^2)-a(b/(2a))^2+c#
= #a(x+b/(2a))^2-b^2/(4a)+c#
= #a(x+b/(2a))^2-(b^2-4ac)/(4a)#
Observe that as #(x+b/(2a))^2# is always greater than #0#,
if #a# is positive, we will have a minima for #y#, when #x+b/(2a)=0# i.e. #x=-b/(2a)#, which will be at #-(b^2-4ac)/(4a)#, and
if #a# is negative, we will have a maxima for #y#, when #x+b/(2a)=0# i.e. #x=-b/(2a)#, which will be at #-(b^2-4ac)/(4a)#.
Hence to find a maxima or minima for a quadratic function, observe the sign of #a# and convert the equation, as above, in form #a(x-h)^2+k#. Then the corresponding maxima or minima will be #k#, when #x=h#.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the maximum or minimum of a quadratic function, you can use the vertex formula: x = -b / (2a). Once you find the x-coordinate of the vertex, substitute it into the quadratic function to find the corresponding y-coordinate. This point represents the maximum or minimum value of the quadratic function, depending on whether the leading coefficient (a) is positive (minimum) or negative (maximum). Alternatively, you can also find the maximum or minimum by completing the square or using the first or second derivative tests.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7