Mathbee CBSE Mathbee · CBSE Class 12 Maths
Class 12 Applications Of Derivatives · Long Answer3 Marksextrema_parameters

Class 12 Maths · Applications Of Derivatives · Long Answer #96

For real a, f(x)=ax²+6x−15 is strictly increasing before 3/4 and strictly decreasing after 3/4. Find the local maximum point of g(x)=ax²−6x+15.

✅ Step-by-Step Solution

Final Answer: (−3/4,69/4)
  1. The derivative of f is 2ax+6. Its sign change at 3/4 requires 2a(3/4)+6=0, giving a=−4.
  2. Then g(x)=−4x²−6x+15 and g′=−8x−6.
  3. g′=0 gives x=−3/4. Since g″=−8<0 it is a local (indeed global) maximum.
  4. g(−3/4)=−9/4+9/2+15=69/4, so the maximum point is (−3/4,69/4).

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] The derivative of f is 2ax+6. Its sign change at 3/4 requires 2a(3/4)+6=0, giving a=−4.
  • [1 mark] Then g(x)=−4x²−6x+15 and g′=−8x−6.
  • [1 mark] g′=0 gives x=−3/4. Since g″=−8<0 it is a local (indeed global) maximum.
  • [0.5 marks] g(−3/4)=−9/4+9/2+15=69/4, so the maximum point is (−3/4,69/4).

Master Chapter 6: Applications Of Derivatives

Practise interactive questions, flashcards, visual mind maps, and timed board tests for Applications Of Derivatives.
📘 Chapter Notes & All Exercises 🚀 Open Interactive Workspace →
← Previous Question
Prove that f(x)=sin x+√3cos x has a maximum at x=π/6, and find that…
Next Question →
Find the maximal intervals of strict increase and decrease for…

More Practice Questions in Applications Of Derivatives

Q1Q2Q3Q4Q5Q6Q7Q8Q9Q10Q11Q12Q13Q14Q15Q16Q17Q18Q19Q20Q21Q22Q23Q24Q25Q26Q27Q28Q29Q30Q31Q32Q33Q34Q35Q36Q37Q38Q39Q40

Related Questions in Applications Of Derivatives

Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximal open interval on which f(x)=x²e^(−x) has positive derivative.
Class 12 Applications Of Derivatives · Mcq Single Correct
For f(x)=x²−4x+6, identify the maximal open interval where f′(x)>0.
Class 12 Applications Of Derivatives · Mcq Single Correct
For x>0, at which x is f(x)=x^x stationary?
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximum value of (1/x)^x for x>0.
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximum of [x(x−1)+1]^(1/3) on 0≤x≤1.
Class 12 Applications Of Derivatives · Mcq Single Correct
A trapezium area model is A(x)=(10+x)√(100−x²) cm² for 0≤x≤10. Find its maximum.