# The minimum value of ${x^2} - 12x + 40$ and the maximum value of $24x - 8 - 9{x^2}$ is?

Last updated date: 20th Mar 2023

•

Total views: 306k

•

Views today: 3.85k

Answer

Verified

306k+ views

Hint-In this question we have to find the minimum and maximum value of the given quadratic equations. Check for the coefficient of highest power term to be positive or negative and apply the respective formula for maximum and minimum value of a quadratic equation.

Complete step-by-step answer:

We have to find the minimum value of ${x^2} - 12x + 40$ and the maximum value of $24x - 8 - 9{x^2}$.

$ \Rightarrow $ Now if we have a quadratic equation of the form ${\text{a}}{{\text{x}}^2} + bx + c = 0$such that $a > 0$ then the minimum value of this quadratic equation is $\dfrac{{4ac - {b^2}}}{{2a}}$at the point x=$\dfrac{b}{{2a}}$ and in this case the maximum value doesn’t exist as the parabola formed is opening upwards and the domain of this equation is R (that is real numbers) thus the max value can go up to infinity.

$ \Rightarrow $ Now if we talk about a quadratic equation of the form ${\text{a}}{{\text{x}}^2} + bx + c = 0$ such that $a < 0$ then the maximum value of this quadratic equation is $\dfrac{{4ac - {b^2}}}{{2a}}$ at point x=$\dfrac{{ - b}}{{2a}}$ and in this case the minimum does not exist as the parabola formed is opening downwards and the domain of this equation is R (that is real numbers) thus the min value can go up to minus infinity.

Now in the equation ${x^2} - 12x + 40$ a=1, b=-12 and c=40

As $a > 0$ because 1>0, hence using the above mentioned concept it does not have a maximum value and the minimum value is given by $\dfrac{{4ac - {b^2}}}{{2a}}$.

So substituting the values we get minimum value = $\dfrac{{4(1) \times 40 - {{\left( { - 12} \right)}^2}}}{{2 \times 1}} = 8$

Now in the equation $24x - 8 - 9{x^2}$, a=-9, b=24, c=-8.

As $a < 0$ because -9<0, hence using the above mentioned concept at does not have a minimum value and the maximum value is given by $\dfrac{{4ac - {b^2}}}{{2a}}$.

So substituting the values we get the maximum value = $\dfrac{{4(9) \times ( - 8) - {{\left( {24} \right)}^2}}}{{2 \times 9}} = 16$

Note – Whenever we face such types of problems the key concept is to understand whether the parabola formed by the equation given is opening up parabola or opening down parabola, it all depends upon the value of the coefficient of highest power of that equation. Then applying the concept mentioned above we can get the answer.

Complete step-by-step answer:

We have to find the minimum value of ${x^2} - 12x + 40$ and the maximum value of $24x - 8 - 9{x^2}$.

$ \Rightarrow $ Now if we have a quadratic equation of the form ${\text{a}}{{\text{x}}^2} + bx + c = 0$such that $a > 0$ then the minimum value of this quadratic equation is $\dfrac{{4ac - {b^2}}}{{2a}}$at the point x=$\dfrac{b}{{2a}}$ and in this case the maximum value doesn’t exist as the parabola formed is opening upwards and the domain of this equation is R (that is real numbers) thus the max value can go up to infinity.

$ \Rightarrow $ Now if we talk about a quadratic equation of the form ${\text{a}}{{\text{x}}^2} + bx + c = 0$ such that $a < 0$ then the maximum value of this quadratic equation is $\dfrac{{4ac - {b^2}}}{{2a}}$ at point x=$\dfrac{{ - b}}{{2a}}$ and in this case the minimum does not exist as the parabola formed is opening downwards and the domain of this equation is R (that is real numbers) thus the min value can go up to minus infinity.

Now in the equation ${x^2} - 12x + 40$ a=1, b=-12 and c=40

As $a > 0$ because 1>0, hence using the above mentioned concept it does not have a maximum value and the minimum value is given by $\dfrac{{4ac - {b^2}}}{{2a}}$.

So substituting the values we get minimum value = $\dfrac{{4(1) \times 40 - {{\left( { - 12} \right)}^2}}}{{2 \times 1}} = 8$

Now in the equation $24x - 8 - 9{x^2}$, a=-9, b=24, c=-8.

As $a < 0$ because -9<0, hence using the above mentioned concept at does not have a minimum value and the maximum value is given by $\dfrac{{4ac - {b^2}}}{{2a}}$.

So substituting the values we get the maximum value = $\dfrac{{4(9) \times ( - 8) - {{\left( {24} \right)}^2}}}{{2 \times 9}} = 16$

Note – Whenever we face such types of problems the key concept is to understand whether the parabola formed by the equation given is opening up parabola or opening down parabola, it all depends upon the value of the coefficient of highest power of that equation. Then applying the concept mentioned above we can get the answer.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE