Q1373: Find the absolute maximum and absolute minimum of function $...

Class CBSE Class 12 Mathematics Applications of Derivatives Q #1373

KNOWLEDGE BASED

UNDERSTAND

5 Marks 2025 AISSCE(Board Exam) LA

Find the absolute maximum and absolute minimum of function $f(x)=2x^{3}-15x^{2}+36x+1$ on $[1, 5]$.

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AI Tutor Explanation

Detailed Solution

Step 1: Find the derivative of the function

To find the absolute maximum and minimum, we first need to find the critical points of the function within the given interval. This involves finding the derivative of the function and setting it equal to zero.

Step 2: Calculate the derivative

Given $f(x) = 2x^3 - 15x^2 + 36x + 1$, we find its derivative $f'(x)$. $$f'(x) = 6x^2 - 30x + 36$$

Step 3: Find the critical points

Set $f'(x) = 0$ and solve for $x$: $$6x^2 - 30x + 36 = 0$$ Divide by 6: $$x^2 - 5x + 6 = 0$$ Factor the quadratic equation: $$(x - 2)(x - 3) = 0$$ So, the critical points are $x = 2$ and $x = 3$.

Step 4: Evaluate the function at critical points and endpoints

Now, we evaluate the function $f(x)$ at the critical points $x = 2$ and $x = 3$, and at the endpoints of the interval $x = 1$ and $x = 5$. $f(1) = 2(1)^3 - 15(1)^2 + 36(1) + 1 = 2 - 15 + 36 + 1 = 24$ $f(2) = 2(2)^3 - 15(2)^2 + 36(2) + 1 = 16 - 60 + 72 + 1 = 29$ $f(3) = 2(3)^3 - 15(3)^2 + 36(3) + 1 = 54 - 135 + 108 + 1 = 28$ $f(5) = 2(5)^3 - 15(5)^2 + 36(5) + 1 = 250 - 375 + 180 + 1 = 56$

Step 5: Determine the absolute maximum and minimum

Comparing the values of $f(x)$ at these points: $f(1) = 24$ $f(2) = 29$ $f(3) = 28$ $f(5) = 56$ The absolute maximum is 56 at $x = 5$, and the absolute minimum is 24 at $x = 1$.

Final Answer: Absolute maximum: 56 at x=5, Absolute minimum: 24 at x=1

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Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the concept of finding absolute maximum and minimum values of a function on a closed interval, which involves finding critical points and evaluating the function at the endpoints.

Knowledge Dimension: PROCEDURAL

Justification: The student needs to apply a specific procedure (finding the derivative, setting it to zero, solving for critical points, and evaluating the function at critical points and endpoints) to solve the problem.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of finding absolute maximum and minimum values of a function, a standard topic in the application of derivatives.