Shayan A.

asked • 04/10/17

Find the set of values of k for which f(x)= 3x^2 -5x -k is greater than 1 for all real x

This is a year 11 methods question about applying function notation.
Could someones please explain the question and then give me a working out for it?

Prentice D.

Ok so this question is asking for what values for k such that the graph never goes below 1. Since this is a hyperbolic function of the form f(x)=ax2 +bx+c (happy face graph due to positive a value), the function will have a lowest point when the derivative of this function is equal to zero. So,
f(x) = 3x2 -5x -k
f'(x) = 6x -5   (notice that the derivative does not depend on k, so the lowest point of the graph will have the same x value no matter the k value).
now sub in f'(x) = 0
0 = 6x-5
x=5/6
 
Now from the original function sub in f(x) = 1, and x=5/6
1 = 3*(5/6)^2 -5(5/6) -k
k=-37/12
k=-3.08 (approximately)
Now the k value will only lower or increase the height of the graph, so increasing the value for k value will put the lowest part on the graph above 1, while decreasing 1 will make the lowest part on the graph less than 1.
So therefore the value for k must be k>-3.08, so that the graph is greater than 1 for all real x.
 
 
Report

04/10/17

Shayan A.

Thank you for your response,
It makes sense using the derivative (I had some personal practice on it before) but we havent officially been taught yet. So I was wondering if there is another way of solving this question.
 
Report

04/10/17

1 Expert Answer

By:

Roman C. answered • 04/10/17

Tutor
5.0 (885)

Masters of Education Graduate with Mathematics Expertise
About this tutor ›
About this tutor ›
The minimum (or maximum) value achieved for y = ax2 + bx + c is y = -Δ/(4a) where Δ = b2 - 4ac is the discriminant.
 
In your case:
 
Δ = b2 - 4ac = (-5)2 - 4(3)(-k) = 25 + 12k
 
ymin = -(25 + 12k)/(4·3) = -25/12 - k
 
-25/12 - k > 1
 
25/12 + k < -1
 
k  < -1 - 25/12
 
k  < -37/12
 
 
Upvote • 0 Downvote
More
Report

Shayan A.

Thank you for your response,
I was just wondering where does the y=-(discriminant)/4a comes from?
Whats the reason that it works?
Report

04/10/17

Still looking for help? Get the right answer, fast.

Ask a question for free

Get a free answer to a quick problem.
Most questions answered within 4 hours.

OR

Find an Online Tutor Now

Choose an expert and meet online. No packages or subscriptions, pay only for the time you need.